Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical...
Transcript of Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical...
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Khovanov homology of framed and signed chord diagrams.
Oleg Viro
December 2, 2006
Knots and links
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
2 / 47
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
That is a circle smoothly embedded into R3 .
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes.
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes.
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes. This gives rise to a knot diagram:
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes. This gives rise to a knot diagram:
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes. This gives rise to a knot diagram:
A link diagram:
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
In many cases 1D picture serves better.
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
In many cases 1D picture serves better.
1D picture comes from a parameterization.
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
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A knot diagram is a 2D picture of knot.
In many cases 1D picture serves better.
1D picture comes from a parameterization.
initial
point
12
3
4
1
23
4
2
1 4
3
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
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Decorate the source:
initial
point
12
3
4
1
23
4
2
1 4
3
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
initial
point
12
3
4
1
23
4
2
1 4
3
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
initial
point
12
3
4
1
23
4
2
1 4
3
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
initial
point
12
3
4
1
23
4
2
1 4
3
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs:
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative .
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative . The result
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative . The result,
1
23
4
2
1 4
3
+−−
+
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative . The result,
1
23
4
2
1 4
3
+−−
+, is called a Gauss diagram of the knot.
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
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Take any such diagram, say,
1
+ +
−32
and try to reconstruct the knot.
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Start with crossings:
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Start with crossings:
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Connect them step by step:
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Connect them step by step:
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
The next step does not work!
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
But let us continue!
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Yet another obstruction!
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
We did it!
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
We did it! But what is the result?
2
3
1
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
We did it! But what is the result?
2
3
1
The result is called a virtual knot diagram.
Virtual links
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
7 / 47
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagram
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?Handles!
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?Handles!
2
3
1
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surface
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.The surfaces is not unique:
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.The surfaces is not unique: one can add handles.
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
10 / 47
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
(R3):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
(R3):
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
(R3):
Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram
(i.e., a plane projection of a link diagram on a surface)
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Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
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Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
Reidemeister moves:
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Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
Reidemeister moves:
Virtual moves:
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Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
Reidemeister moves:
Virtual moves:
All virtual moves can be replaced by detour moves:
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Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Gauss diagrams has nothing to do with virtual crossings!
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Gauss diagrams has nothing to do with virtual crossings!They do not change under virtual moves.
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
Nega-tive firstmove
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
+
Nega-tive firstmove
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
+
Nega-tive firstmove
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
+
Nega-tive firstmove
−
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
Thirdmove
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
−ε
ε
Thirdmove
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
−ε
ε
Thirdmove
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
−ε
ε
Thirdmove β
γ
α
β
γ
α
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagrams
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagramsIsotopies → Reidemeister moves
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagramsIsotopies → Reidemeister moves
Combinatorial incarnations of virtual knot theory
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagramsIsotopies → Reidemeister moves
Combinatorial incarnations of virtual knot theory
GaussDiagrams
←Virtual Links(?)
→Virtual LinkDiagrams
ReidemeisterMoves
←VirtualIstopies (?)
→Reidemeisterand Detourmoves
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Bridges combinatorics
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Bridges combinatorics (= 1D topology)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Bridges combinatorics with (3D-) topology.
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem:
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group,the fundamental group of the link complement R
3r link .
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized.
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized, even in two ways!
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!In terms of links in a thickened surface this is the fundamentalgroup of the complement, but with one of two sides of theboundary contracted to a point.
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!In terms of links in a thickened surface this is the fundamentalgroup of the complement, but with one of two sides of theboundary contracted to a point.
Kauffman bracket is more practical and elementary invariant.
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
17 / 47
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
(a Laurent polynomial in A with integer coefficients).
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 =
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈 〉 = A〈 〉+ A−1〈 〉 (Kauffman Skein Relation) .
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈 〉 = A〈 〉+ A−1〈 〉 (Kauffman Skein Relation) .Uniqueness is obvious.
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈 〉 = A〈 〉+ A−1〈 〉 (Kauffman Skein Relation) .Uniqueness is obvious.Invariant under R2 and R3, under R1 multiplies by −A±3.
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram:
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
,
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, ,
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , ,
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , , ,
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , , , . . .
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , , , . . .
Totally 2c states, where c is the number of crossings.
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→ smoothing(s) =
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→ smoothing(s) =
|s| = 2
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→ smoothing(s) =
|s| = 2
State Sum: 〈D〉 =∑
s state of D Aa(s)−b(s)(−A2 − A−2)|s|
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
=
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
=
A2(−A2−A−2)2 + 2(−A2−A−2) + A−2(−A2−A−2)2 =
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
=
A2(−A2−A−2)2 + 2(−A2−A−2) + A−2(−A2−A−2)2 =
A6 + A2 + A−2 + A−6
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Smoothing depends only of the signs of marker and crossing.
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Smoothing depends only of the signs of marker and crossing.No need in direction of the arrow!
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Smoothing depends only of the signs of marker and crossing.No need in direction of the arrow!Kauffman state sum is defined for signed chord diagrams.
Gauss diagrams of a poorman
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
23 / 47
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), and
B
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and• each chord is equipped with a sign
−+
−−−
+ +
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and• each chord is equipped with a signis called a signed chord diagram.
−+
−−−
+ +
State of signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
25 / 47
A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.
State of signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
25 / 47
A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.These are marker signs,
State of signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
25 / 47
A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.These are marker signs, the original signs are structure signs.
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.If σ = + , the Morse modification preserves the structureorientation.
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.If σ = + , the Morse modification preserves the structureorientation.If σ = − , the Morse modification destroys the orientation.
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
+
−
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
If the link is not oriented, specify the framing on the chordsgiving positive smoothing.
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
If the link is not oriented, specify the framing on the chordsgiving positive smoothing.
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
If the link is not oriented, specify the framing on the chordsgiving positive smoothing.
shorthand notation
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framing
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framing
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
Kauffman bracket state sum is defined for a framed chorddiagram.
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
−−
+
−
+−
+
Kauffman bracket state sum is defined for a framed chorddiagram.A state is a distribution of signs over the set of chords.
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
−−
+
−
+−
+
Kauffman bracket state sum is defined for a framed chorddiagram.A state is a distribution of signs over the set of chords.The smoothing defined by a state is according to the famingalong the chords marked with + and the opposite oneotherwise.
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation,on a chord with − take the framing surgery along whichreverses the orientation.
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation,on a chord with − take the framing surgery along whichreverses the orientation.Forget the orientation.
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R
3 .
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R
3 .
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R
3 .
A neighborhood of M in R3 is orientable and fibers over M .
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
2. Cut S along C : S 7→ S C
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
2. Cut S along C : S 7→ S C
3. Thicken: (S C)× R .
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
2. Cut S along C : S 7→ S C
3. Thicken: (S C)× R .4. Paste over the sides of the cut (x+, t) ∼ (x−,−t) .
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.(together with moves)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.Corollary (Bourgoin). Links in orientable thickenings ofsurfaces have well-defined Kauffman bracket.
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
33 / 47
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) .
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Hp,q(D) = Hw(D)−q−2p
2,3w(D)−q
2 (D)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Hp,q(D) = Hw(D)−q−2p
2,3w(D)−q
2 (D) , or
Hi,j(D) = Hj−i−w(D),3w(D)−2j(D).
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Hp,q(D) = Hw(D)−q−2p
2,3w(D)−q
2 (D) , or
Hi,j(D) = Hj−i−w(D),3w(D)−2j(D).
In other words: Hp,q(D) = Hi,j(D) iff
q + 2j = 3w(D) and j − i + p = w(D) .
Enhanced states
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
35 / 47
Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.
Enhanced states
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
35 / 47
Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.
For example: state with smoothing
gives rise to 4 enhanced states
Enhanced states
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
35 / 47
Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.
For example: state with smoothing
gives rise to 4 enhanced states
−
+
+
+
+
−
−
−
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Invariance of Hp,q(D) under Reidemeister moves wanted!
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Invariance of Hp,q(D) under Reidemeister moves wanted!
∂(S) =∑
±T with T , which differ from S by a singlemarker and appropriate signs on the circles passing near thevertex.
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Invariance of Hp,q(D) under Reidemeister moves wanted!
∂(S) =∑
±T with T , which differ from S by a singlemarker and appropriate signs on the circles passing near thevertex.
(|T | − |S|) = 1 is needed to have τ(T ) = τ(S)− 1 .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .
Then⊕p,qCp,q(D) = ⊕sVs
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .
Then⊕p,qCp,q(D) = ⊕sVs
Differentials are defined by the multiplication andco-multiplication in A .
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
This works for classical links, but does not for virtual!
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
This works for classical links, but does not for virtual!
For virtual links, it works with Z2 coefficients.
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
− A4 − 1 −−−→ −A2 − A−2
y
y
A4 + 2 + A−4 −−−→ −1− A−4
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
− A4 − 1 −−−→ −A2 − A−2
y
y
A4 + 2 + A−4 −−−→ −1− A−4
Differentials are obvious in all A -components but the onecorresponding to A0 .
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
− A4 − 10
−−−→ −A2 − A−2
y0
y
A4 + 2 + A−4 −−−→ −1− A−4
Differentials are obvious in all A -components but the onecorresponding to A0 .
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
10
−−−→
y0
y
1⊗X + X ⊗ 1 −−−→ 2× 1
Differentials are obvious in all A -components but the onecorresponding to A0 .
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
10
−−−→
y0
y
1⊗X + X ⊗ 1 −−−→ 2× 1
This does not happen if the chord diagram is orientable!
Orientation of chorddiagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
39 / 47
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientability of chord diagram with connected base is
equivalent to the following condition known to K.-F.Gauss:
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientability of chord diagram with connected base is
equivalent to the following condition known to K.-F.Gauss:
The number of endpoints of chords on each arc bounded be
endpoints of a chord is even.
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientability of chord diagram with connected base is
equivalent to the following condition known to K.-F.Gauss:
The number of endpoints of chords on each arc bounded be
endpoints of a chord is even.
The simplest nonorientable chord diagram: ⊗ .
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
The obstruction to orientability of a chord diagram
(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
The obstruction to orientability of a chord diagram
(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .
Dual class belongs to H0(B r ∪ni=1∂ci; Z2) .
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
The obstruction to orientability of a chord diagram
(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .
Dual class belongs to H0(B r ∪ni=1∂ci; Z2) .
Orient the complement of the 0-cycle realizing it,
to get vice-orientation of the chord diagram.
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram
+ +
(−)
(+)
(+) (−)
−
−
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
its orientation induces an orientation of each result of itssmoothing.
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(−)
−
−
(+)
its orientation induces an orientation of each result of itssmoothing.
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(−)
−
−
(+)
its orientation induces an orientation of each result of itssmoothing.Similarly, a vice-orientation of a signed chord diagram inducesa vice-orientation of each result of its smoothing.
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(−)
−
−
(+)
its orientation induces an orientation of each result of itssmoothing.Similarly, a vice-orientation of a signed chord diagram inducesa vice-orientation of each result of its smoothing.
Theorem (Manturov, Viro) Definition of the Khovanov complexextended straightforwardly to an oriented framed chorddiagram gives a complex invariant under Reidemeister movespreserving the orientation.
Khovanov complex offramed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
43 / 47
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphisms
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphismsneeded for construction of differentials.
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphismsneeded for construction of differentials.Homology does not depend on the structure.
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .But ∆(conj(1)) = ∆(1) = X ⊗ 1 + 1⊗X
= −∆(X)⊗∆(1)−∆(1)⊗∆(X) .
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .But ∆(conj(1)) = ∆(1) = X ⊗ 1 + 1⊗X
= −∆(X)⊗∆(1)−∆(1)⊗∆(X) .and∆(conj(X)) = ∆(−X) = −X ⊗X = −∆(X)⊗∆(X) .
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:Reversing of orientation of a component corresponds to conjin the corresponding copy of A .
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:Reversing of orientation of a component corresponds to conjin the corresponding copy of A .Permutations of the components corresponds to thepermutation isomorphism of the tensor product multiplied bythe sign of the permutation.
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.Orient the first components according to thevice orientation at c .
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.Orient the first components according to thevice orientation at c . In these representations of Vs and Vt ,define the map by multiplication or co-multipication.