Introduction instanton molecules and topological susceptibility Random matrix model
Topological Symmetries of Molecules · Topological Symmetries of Molecules Erica Flapan December...
-
Upload
nguyentruc -
Category
Documents
-
view
218 -
download
0
Transcript of Topological Symmetries of Molecules · Topological Symmetries of Molecules Erica Flapan December...
Topological Symmetries of Molecules
Erica Flapan
December 13, 2013
Workshop: Topological Structures in Computational BiologyInstitute for Mathematics and its Applications
Erica Flapan Topological Symmetries of Molecules
Molecular symmetries
The symmetries of a molecule determine many important aspectsof its behavior.
For example, symmetry is useful for:
• Predicting reactions
• Crystallography
• Spectroscopy
• Quantum chemistry
• Analyzing the electron structure of a molecule
• Classifying molecules
Erica Flapan Topological Symmetries of Molecules
Living organisms
Asymmetric molecules interact with one another like feet andshoes.
Asymmetric objects
Erica Flapan Topological Symmetries of Molecules
Living organisms
Asymmetric molecules interact with one another like feet andshoes.
Asymmetric objects
Amino acids, sugars, and other molecules in living organisms areasymmetric.
DNA is different from
its mirror image
Hence we react differently to mirror forms of asymmetric molecules.
Erica Flapan Topological Symmetries of Molecules
Pharmaceuticals
Some pharmaceuticals and their mirror images:
• Ibuprofen is an anti-inflamatory, but its mirror form is inert.
• Naproxen is an anti-inflamatory, but its mirror form is toxic.
• Darvon is a pain killer, but its mirror form is the coughsuppressant Novrad.
Erica Flapan Topological Symmetries of Molecules
Pharmaceuticals
Some pharmaceuticals and their mirror images:
• Ibuprofen is an anti-inflamatory, but its mirror form is inert.
• Naproxen is an anti-inflamatory, but its mirror form is toxic.
• Darvon is a pain killer, but its mirror form is the coughsuppressant Novrad.
Drugs are synthesized in a 50:50 mix of mirror forms.
If a molecule has mirror image symmetry these are the same.
Otherwise, the two forms may need to be separated to avoiddangerous side effects.
Knowing whether a structure will have mirror symmetry is useful indrug design.
Erica Flapan Topological Symmetries of Molecules
Mirror image symmetry
But what do we mean by mirror symmetry?.
Definition:
A molecule is said to be chemically chiral if it can not transformitself into its mirror image at room temperature. Otherwise, it issaid to be chemically achiral.
Note: This definition describes the behavior of a molecule not itstopology or geometry.
Erica Flapan Topological Symmetries of Molecules
Mirror image symmetry
But what do we mean by mirror symmetry?.
Definition:
A molecule is said to be chemically chiral if it can not transformitself into its mirror image at room temperature. Otherwise, it issaid to be chemically achiral.
Note: This definition describes the behavior of a molecule not itstopology or geometry.
Definition:
A rigid object is said to be geometrically chiral if it cannot besuperimposed on its mirror image. Otherwise, it is said to begeometrically achiral.
Erica Flapan Topological Symmetries of Molecules
Geometric vs.chemical achirality
If an object can be rigidly superimposed on its mirror image, thenit is chemically the same as it’s mirror image.
Geometrically
AchiralChemically
Achiral(the same as mirror
image as a rigid object)
(can transform itself
into its mirror image)
Erica Flapan Topological Symmetries of Molecules
Geometric vs.chemical achirality
If an object can be rigidly superimposed on its mirror image, thenit is chemically the same as it’s mirror image.
Geometrically
AchiralChemically
Achiral(the same as mirror
image as a rigid object)
(can transform itself
into its mirror image)
Thus the set of geometrically achiral molecules is a subset of theset of chemically achiral molecules.
geometrically achiral
chemically achiral
? But is there a chemically achiral molecule
which is not geometrically achiral?
C
H
CH3
H
C
H3C
Cl Cl
Erica Flapan Topological Symmetries of Molecules
Geometric vs.chemical chirality
This molecule (without the faces) was synthesized by Kurt Mislowin 1954 to show that geometric and chemical chirality are different.
NO2
O
NO2
O2N
OO2N
O
C C
O
left propeller right propeller
{ {Propellers are behind the screen. They turn simultaneously.
Erica Flapan Topological Symmetries of Molecules
Geometric vs.chemical chirality
This molecule (without the faces) was synthesized by Kurt Mislowin 1954 to show that geometric and chemical chirality are different.
NO2
O
NO2
O2N
OO2N
O
C C
O
left propeller right propeller
{ {Propellers are behind the screen. They turn simultaneously.
Left propeller has her left hand forward, right propeller has herright hand forward. So, as rigid structures, a right propeller isdifferent from a left propeller.
Erica Flapan Topological Symmetries of Molecules
Molecule is chemically achiral
NO
NO2
2
O2N
O2N
O
O
C
O
C
O
NO2
O
NO2
O2N
OO2N
O
C C
O
Original Mirror form
Erica Flapan Topological Symmetries of Molecules
Molecule is chemically achiral
NO
NO2
2
O2N
O2N
O
O
C
O
C
O
NO2
O
NO2
O2N
OO2N
O
C C
O
Original Mirror form
Mirror form is the same as original, except vertical and horizontalhexagons have switched places.
Erica Flapan Topological Symmetries of Molecules
Molecule is chemically achiral
NO
NO2
2
O2N
O2N
O
O
C
O
C
O
NO2
O
NO2
O2N
OO2N
O
C C
O
Original Mirror form
Mirror form is the same as original, except vertical and horizontalhexagons have switched places.
Proof of chemical achirality
Erica Flapan Topological Symmetries of Molecules
Molecule is chemically achiral
NO
NO2
2
O2N
O2N
O
O
C
O
C
O
NO2
O
NO2
O2N
OO2N
O
C C
O
Original Mirror form
Mirror form is the same as original, except vertical and horizontalhexagons have switched places.
Proof of chemical achirality
• Rotate original molecule by 90◦ about a horizontal axis to getmirror form with propellers horizontal.
Erica Flapan Topological Symmetries of Molecules
Molecule is chemically achiral
NO
NO2
2
O2N
O2N
O
O
C
O
C
O
NO2
O
NO2
O2N
OO2N
O
C C
O
Original Mirror form
Mirror form is the same as original, except vertical and horizontalhexagons have switched places.
Proof of chemical achirality
• Rotate original molecule by 90◦ about a horizontal axis to getmirror form with propellers horizontal.
• Rotate propellers back to vertical position to get mirror form.
Erica Flapan Topological Symmetries of Molecules
Molecule is geometrically chiral
NO
NO2
2
O2N
O2N
O
O
C
O
C
O
NO2
O
NO2
O2N
OO2N
O
C C
O
Original Mirror form
Proof of geometric chirality
• Suppose molecule is rigid. Then propellers don’t rotate.
• In original form, left propeller is parallel to adjacent hexagon.
• In mirror form, left propeller is perpendicular to adjacenthexagon.
• A left propeller cannot change into a right propeller.
• As rigid objects, the original and mirror form are distinct.
Erica Flapan Topological Symmetries of Molecules
Geometric vs.chemical chirality
This example shows that for non-rigid molecules, geometricchirality does not necessarily imply chemical chirality.
chemically achiral
NO2
O
NO2
O2N
OO2N
O
C C
O
geometrically
achiralC
H
CH3
H
C
H3C
Cl Cl
Erica Flapan Topological Symmetries of Molecules
Molecular rigidity and non-rigidity
In fact, some molecules are rigid, some are flexible, and some havepieces that can rotate around certain bonds.
O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
C
H
H
H
H
rigid flexible
Br
Cl
ClCl
H C3
rotating propeller
NO2
O
NO2
O2N
O
O2N
O
C C
O
two simultaneous propellersErica Flapan Topological Symmetries of Molecules
Topological Chirality
So no mathematical characterization of chemical chirality thatworks for all molecules is possible.
The definition of geometric chirality treats all molecules ascompletely rigid, which is not correct.
Now we will treat all molecules as completely flexible, which is alsonot correct.
The truth is somewhere in the middle.
Definition
A molecule is said to be topologically achiral if, assuming completeflexibility, it is isotopic to its mirror image. Otherwise it is said tobe topologically chiral.
Erica Flapan Topological Symmetries of Molecules
Topological vs chemical chirality
If this molecule were flexible, we could grab the CO2H and push itto the left, while pulling the H to the right.
C
NH2
H
CH3
CO2H H
CH3
C
NH2
HO2C
mirrorinterchange
So it is topologically achiral. However, the molecule is rigid so it’schemically and geometrically chiral.
Topologically
achiralChemically
achiral
(the same as mirror image
as a flexible object)
(the same as mirror
image experimentally)
Erica Flapan Topological Symmetries of Molecules
Topological chirality
Topologically chiralGeometrically chiral
(different from mirror
image as a rigid
object)
(different from mirror
image experimentally)
(different from mirror
image as a flexible
object)
Chemically chiral
None of the reverse implications hold.
Erica Flapan Topological Symmetries of Molecules
Topological chirality
Topologically chiralGeometrically chiral
(different from mirror
image as a rigid
object)
(different from mirror
image experimentally)
(different from mirror
image as a flexible
object)
Chemically chiral
None of the reverse implications hold.
If we heat a molecule which is geometrically chiral but nottopologically chiral, we can force it to change to its mirror form.
Even if we heat a topologically chiral molecule, it will not changeto its mirror form.
Thus knowing whether or not a molecule is topologically chiralhelps to predict its behavior.
Erica Flapan Topological Symmetries of Molecules
The first example
In 1986, Jon Simon gave the first example of a topologically chiralmolecule by proving that a molecular Mobius ladder with threerungs is topologically chiral.
O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
We sketch Simon’s proof.
Erica Flapan Topological Symmetries of Molecules
Set-up
We represent the molecule as a colored graph M3, distinguishingsides from the rungs, because chemically they are different.
O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
M3
The different rung colors help us keep track of each rung, but arenot meant to distinguish one from another.
Erica Flapan Topological Symmetries of Molecules
Set-up
We represent the molecule as a colored graph M3, distinguishingsides from the rungs, because chemically they are different.
O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
M3
The different rung colors help us keep track of each rung, but arenot meant to distinguish one from another.
Isotop sides of ladder to a planar circle A.
AErica Flapan Topological Symmetries of Molecules
2-fold branched covers
We obtain the 2-fold branched cover, by gluing two copies ofladder together along A.
A= p(fix(h))h fix(h)
p
S3M= S
3N=
quotient map
Formal definition
M, N = 3–manifolds, h : M → M orientation preservinghomeomorphism of order 2, and p : M → N quotient map inducedby h. If A = p(fix(h)) is a 1–manifold, then we say M is the 2–foldbranched cover of N branched over A.
Erica Flapan Topological Symmetries of Molecules
Sketch of branched cover argument
Remove A to obtain a link L.
2-fold branched cover L
Use linking numbers to prove that the link L is topologically chiral.
If the Mobius ladder M3 were topologically achiral, then we couldlift the isotopy to get an isotopy taking L to its mirror image.
Thus the molecular Mobius ladder is topologically chiral,distinguishing the rungs from the sides.
Erica Flapan Topological Symmetries of Molecules
Mobius strip
Maybe this is not surprising, since a Mobius strip is topologicallychiral.
mirror
Erica Flapan Topological Symmetries of Molecules
Another Mobius ladder
Kuratowski cyclophane also has the underlying form of a molecularMobius ladder M3, though its molecular structure is quite different.
O
O O
O
O
O
O
O
O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
Kuratowski cylcophane
Mobius ladderembedded
graph
Erica Flapan Topological Symmetries of Molecules
An achiral Mobius ladder with three rungs
O
O O
O
O
O
O
O
O
O O
O
O
O
O
O
The black is in the plane of reflection, pink is in front, and blue isin back.
Thus Kuratowski cyclophane is geometrically achiral, though hasthe form of a Mobius ladder with three rungs.
Erica Flapan Topological Symmetries of Molecules
A Mobius ladder with four rungs
We see as follows that this iron-sulfur cluster contains a Mobiusladder with four rungs.
[CH2]8
[CH2]8
[CH2]8
[CH2]8
N
N
N
N
R
S
Fe
R
S
Fe
S
S
SR
S
Fe
S
R
Fe
S
3
4
5
8
1
2
6
7
underlying structure
Erica Flapan Topological Symmetries of Molecules
Mobius ladder with four rungs
The sides of the Mobius ladder are green and the rungs are pink.We ignore the black.
3
4
5
8
1
2
6
7
1
2
3
4
5
6
7
8
Erica Flapan Topological Symmetries of Molecules
Structure is geometrically achiral
[CH2]8
[CH2]8
[CH2]8
NN
NN
R
SFe
R
S
Fe
S
S
S
R
S
Fe
SR
Fe
S
[CH2]8
[CH2]8
[CH2]8
[CH2]8
N
N
N
N
R
S
Fe
R
S
Fe
S
S
SR
S
Fe
S
R
Fe
S
mirror
rotate clockwise
[CH2]8
To get mirror image, rotate clockwise by 90◦.Erica Flapan Topological Symmetries of Molecules
Topological Chirality
Various methods have been developed to prove that graphsembedded in R3 are topologically chiral, whether or not they aremolecular graphs.
Theorem [Flapan]
If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then any embedding of the graph in R3 istopologically chiral.
a b c
1 2 3K3,3
1
2 3
4
5
K5
Erica Flapan Topological Symmetries of Molecules
Ferrocenophane
O
Feferrocenophane
fixed2
4
Any automorphism of a molecular graph must take atoms of agiven type to atoms of the same type.
Hence any automorphism of ferrocenophane fixes the oxygen, andhence fixes the adjacent vertex.
Since a automorphism cannot interchange vertices of differentvalence, it must also fix vertex 2 and vertex 4.
Erica Flapan Topological Symmetries of Molecules
Ferrocenophane
O
Feferrocenophane
fixed
fixed
fixed
Now progressively we see that more and more vertices are fixed.
Erica Flapan Topological Symmetries of Molecules
Ferrocenophane
O
Feferrocenophane
fixed
fixed
fixed
Now progressively we see that more and more vertices are fixed.
O
Feferrocenophane
fixed
fixed
fixed
fixedfixed
fixed
In fact, every vertex is fixed. So ferrocenophane has no non-trivialautomorphisms.
Erica Flapan Topological Symmetries of Molecules
Ferrocenophane
To see ferrocenophane contains K5.
12
3
4
5
O
Fe
12
3
4
5K 5
Erica Flapan Topological Symmetries of Molecules
Ferrocenophane
To see ferrocenophane contains K5.
12
3
4
5
O
Fe
12
3
4
5K 5
Thus, since it has no order 2 automorphism and contains K5,ferrocenophane is topologically chiral by the theorem.
Erica Flapan Topological Symmetries of Molecules
Intrinsic chirality
Definition
A graph G is said to be intrinsically chiral, if every embedding of Gin space is topologically chiral.
That is, the chirality is intrinsic to the graph and does not dependon the particular embedding.
Erica Flapan Topological Symmetries of Molecules
Intrinsic chirality
Definition
A graph G is said to be intrinsically chiral, if every embedding of Gin space is topologically chiral.
That is, the chirality is intrinsic to the graph and does not dependon the particular embedding.
Theorem [Flapan]
If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then any embedding of the graph in R3 istopologically chiral.
Theorem Restatement
If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then the graph is intrinsically chiral.
Erica Flapan Topological Symmetries of Molecules
Intrinsically chirality
Theorem Restatement
If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then the graph is intrinsically chiral.
In chemical terms, if a molecule is intrinsically chiral then it and allof its stereoisomers are topologically chiral.
O
Feferrocenophane
Thus ferrocenophane is intrinsically chiral.
Erica Flapan Topological Symmetries of Molecules
Topological chirality does not imply intrinsically chirality
These are different embeddings of the same molecular graph.
O
OO O O
O
O
OO
O O OO
O
N N N N
N N N N
OO
OO
O
O
O
O
O
OO O
O
O
N
N
N
N
N
N
N
N
Erica Flapan Topological Symmetries of Molecules
Topological chirality does not imply intrinsically chirality
These are different embeddings of the same molecular graph.
O
OO O O
O
O
OO
O O OO
O
N N N N
N N N N
OO
OO
O
O
O
O
O
OO O
O
O
N
N
N
N
N
N
N
N
The knotted molecule is topologically chiral (because a trefoil knotis topologically chiral), and the unknotted molecule is topologicallyachiral (because it’s planar).
Thus the knotted molecule is topologically chiral but notintrinsically chiral.
Erica Flapan Topological Symmetries of Molecules
Hierarchy of chirality
Topologically chiral
Geometrically chiral
(different from mirror
image as a rigid
object)
(different from mirror
image experimentally)
(different from mirror
image as a flexible
object)Chemically chiral
Intrinsically chiral
(all embeddings different
from mirror image as a
flexible object)
O
OO
O OO
O
O OO O O
O
ON N
H
CH3
C
NH2
HO2C
NO2
O
NO2
O2N
OO2N
O
C C
O
ON N
N NN N
Fe
None of the reverse implications hold.Erica Flapan Topological Symmetries of Molecules
Other types of symmetries
Mirror image symmetry is not the only type of molecular symmetrythat is chemically significant.
Chemists define
The point group of a molecular graph as its group of rotations,reflections, and reflections composed with rotations.
It’s called the point group because it fixes a point of R3.
Chemists use the point group to classify molecules.
Erica Flapan Topological Symmetries of Molecules
Non-rigid molecules
Like geometry chirality, the point group treats all molecules as ifthey are rigid. But as we saw, not all molecules are rigid.
The top of this molecule spins like a propeller.
Br
Cl
Cl
Cl
H C3
Erica Flapan Topological Symmetries of Molecules
Non-rigid molecules
Like geometry chirality, the point group treats all molecules as ifthey are rigid. But as we saw, not all molecules are rigid.
The top of this molecule spins like a propeller.
Br
Cl
Cl
Cl
H C3
Planar reflection pointwise fixing the three hexagons is its onlyrigid symmetry. So point group is Z2.
We would like a symmetry group that includes the reflection aswell as an order 3 rotation of the propeller.
Erica Flapan Topological Symmetries of Molecules
Molecular symmetry group
Definition
Let Γ be a molecular graph, and let Aut(Γ) denote the group ofautomorphisms of Γ taking atoms of a given type to atoms of thesame type. The molecular symmetry group of Γ is the subgroup ofAut(Γ) induced by chemically possible motions taking Γ to itself orits reflection.
Erica Flapan Topological Symmetries of Molecules
Molecular symmetry group
Definition
Let Γ be a molecular graph, and let Aut(Γ) denote the group ofautomorphisms of Γ taking atoms of a given type to atoms of thesame type. The molecular symmetry group of Γ is the subgroup ofAut(Γ) induced by chemically possible motions taking Γ to itself orits reflection.
Cl Br
12
3
The molecular symmetry group is 〈(12), (123)〉.
Erica Flapan Topological Symmetries of Molecules
Molecular symmetry group
Cl Br
1
2
3
circle
Molecular symmetry group induces an isomorphic action on thecircle at the top.
1
2
3
D = <(123),(23)>=
dihedral group with 6 elements3
Molecular symmetry group = D3 = Aut(Γ)
Erica Flapan Topological Symmetries of Molecules
Geometry vs topology
Definition
The topological symmetry group TSG(Γ) of a molecular graph Γ,is the subgroup of Aut(Γ) induced by homeomorphisms of R3
taking atoms of a given type to atoms of the same type.
Analogous to what we saw with achirality
• The point group treats all molecules as completely rigid.
• The topological symmetry group treats all molecules ascompletely flexible.
• The truth is somewhere in the middle.
Erica Flapan Topological Symmetries of Molecules
Topological symmetry groups
Cl Br
1
2
3
circle
For this molecule, TSG(Γ) = molecular symmetry group = D3.
Point group 6=molecular symmetry group.
Erica Flapan Topological Symmetries of Molecules
Topological symmetry groups
Cl Br
1
2
3
circle
For this molecule, TSG(Γ) = molecular symmetry group = D3.
Point group 6=molecular symmetry group.
Another example:
O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
Erica Flapan Topological Symmetries of Molecules
Molecular Mobius Ladder
We represent molecule as a colored graph where automorphismsmust preserve colors.
1
2
3
4
5
6O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
(23)(56)(14) is the only automorphism induced by a rigid motion.So point group = Z2.
Erica Flapan Topological Symmetries of Molecules
Molecular Mobius Ladder
We represent molecule as a colored graph where automorphismsmust preserve colors.
1
2
3
4
5
6O
O
O OO
O
O
O
OO
O
O
OO
O
O
O
O
(23)(56)(14) is the only automorphism induced by a rigid motion.So point group = Z2.
(123456) is induced by rotating the molecule by 120◦ whileslithering the half-twist back to its original position.
TSG(Γ) = Molecular symmetry group = 〈(23)(56)(14), (123456)〉
So point group 6= Molecular symmetry group.
Erica Flapan Topological Symmetries of Molecules
Different types of symmetry groups
⊆⊆ ⊆Point
Group
Molecular
Symmetry
Group
Topological
Symmetry
Group
Automorphism
Group
automorphisms
of the abstract
graph
automorphisms
induced by
molecular
motions
automorphisms
induced by
rotations
and
reflections
automorphisms
induced by
homeomorphisms
of space
Note
The point group is normally defined in terms of rotations andreflections of R3 rather than in terms of automorphisms. We writeit this way to compare it to the other types of symmetry groups.
Erica Flapan Topological Symmetries of Molecules
Arbitrary Graphs embedded in S3
While motivated by molecular symmetries, TSG(Γ) can be definedfor any graph Γ embedded in R3.
Embedded graphs are a natural extension of knot theory, since wecan put vertices on a knot to make it into an embedded graph.
Symmetries are nicer in S3 = R3 ∪ {∞} than in R3.
1 2
The ends of this knot are attached in S3 and (12) is induced by arotation-reflection. If the ends are attached in R3, the symmetry isnot so nice.
1 2
Erica Flapan Topological Symmetries of Molecules
Topological Symmetry Groups
Definition
The topological symmetry group of a graph Γ embedded in S3,TSG(Γ), is the subgroup of Aut(Γ) induced by homeomorphismsof (S3, Γ).
Erica Flapan Topological Symmetries of Molecules
Topological Symmetry Groups
Definition
The topological symmetry group of a graph Γ embedded in S3,TSG(Γ), is the subgroup of Aut(Γ) induced by homeomorphismsof (S3, Γ).
Frucht proved that every finite group is isomorphic to Aut(Γ) forsome graph Γ.
Is every finite group isomorphic to TSG(Γ) for some graph Γembedded in S3?
Before answering, we illustrate some groups that are topologicalsymmetry groups.
Erica Flapan Topological Symmetries of Molecules
What groups can be TSG(Γ)?
Γ
chiral
Wheels can rotate but can’t be interchanged.
TSG(Γ) = Z2 × Z3 × Z4
Erica Flapan Topological Symmetries of Molecules
Any finite abelian group
We can have any number of wheels with any number of spokes.
If two wheels have the same number of spokes, we can add distinct(chiral) knots so the wheels can’t be interchanged.
Erica Flapan Topological Symmetries of Molecules
Any finite abelian group
We can have any number of wheels with any number of spokes.
If two wheels have the same number of spokes, we can add distinct(chiral) knots so the wheels can’t be interchanged.
Γ
TSG(Γ) = Z2 × Z2 × Z4
In this way, any finite abelian group can be TSG(Γ).
Erica Flapan Topological Symmetries of Molecules
Symmetric groups
v
w
1 2 n
non-invertible
chiral
w
v
w
vrotated flipped
over
A non-invertible knot is one that is not isotopic to itself with itsorientation reversed.
Erica Flapan Topological Symmetries of Molecules
Symmetric groups
v
w
1 2 n
non-invertible
chiral
w
v
w
vrotated flipped
over
A non-invertible knot is one that is not isotopic to itself with itsorientation reversed.
Non-invertible & chiral knots ⇒ no homeomorphism induces (vw).
Any transposition (ij) is induced by twisting strands.
Thus TSG(Γ) = Sn, the group of permutations of n points.
Erica Flapan Topological Symmetries of Molecules
What about alternating groups?
v
w
1 2 n
non-invertible
chiral
Can we get TSG(Γ) = An by adding different knots?
Can we get TSG(Γ) = An for another embedded graph Γ?
Erica Flapan Topological Symmetries of Molecules
What about alternating groups?
v
w
1 2 n
non-invertible
chiral
Can we get TSG(Γ) = An by adding different knots?
Can we get TSG(Γ) = An for another embedded graph Γ?
Not unless n ≤ 5.
Theorem [Flapan, Naimi, Pommersheim, Tamvakis]
TSG(Γ) can be An for some graph Γ embedded in S3 iff n ≤ 5.
Erica Flapan Topological Symmetries of Molecules
TSG+(Γ)
Definition
TSG+(Γ) is the subgroup of TSG(Γ) induced by orientationpreserving homeomorphisms of S3.
TSG +(Γ)= Ζ2× Ζ3
× Ζ4TSG +
(Γ)= Ζ2× Ζ3
× Ζ4
TSG (Γ)= Ζ2× Ζ3
× Ζ4( ) Ζ2
TSG (Γ)= Ζ2× Ζ3
× Ζ4
Erica Flapan Topological Symmetries of Molecules
Finite order homeomorphisms
TSG+(Γ) = either TSG(Γ) or a normal subgroup of index 2.
So studying TSG+(Γ) is almost as good as studying TSG(Γ), butit’s simpler.
Erica Flapan Topological Symmetries of Molecules
Finite order homeomorphisms
TSG+(Γ) = either TSG(Γ) or a normal subgroup of index 2.
So studying TSG+(Γ) is almost as good as studying TSG(Γ), butit’s simpler.
A function f has finite order if for some n > 0, f n is the identity.
All automorphisms in TSG+(Γ) have finite order, but are theyinduced by finite order homeomorphisms of S3?
Consider what happens to the red arc when we spin a wheel.
12
3
1
2
3
1
23
12
3
Erica Flapan Topological Symmetries of Molecules
Homeomorphisms of (S3, Γ) may not have finite order.
The wheel returns to its original position, but the red arc does not.
12
3
1
2
3
1
23
12
3
Spinning a wheel has finite order on Γ, but not on S3.
Г
TSG+(Γ) is not induced by a finite group of homeomorphisms of(S3, Γ).
But, this is a special (bad) type of graph.
Erica Flapan Topological Symmetries of Molecules
3-connected graphs
Definition
An abstract graph γ is 3-connected if at least 3 vertices togetherwith their edges must be removed in order to disconnect γ orreduce it to a single vertex.
Erica Flapan Topological Symmetries of Molecules
3-connected graphs
Definition
An abstract graph γ is 3-connected if at least 3 vertices togetherwith their edges must be removed in order to disconnect γ orreduce it to a single vertex.
v
w
1 2 n
removered verticesto disconnectgraphs
Neither of these graphs is 3-connected.
Erica Flapan Topological Symmetries of Molecules
A 3-connected graph
1
3
4
5 62
3-connected
(56)(23) is induced by turning the graph over.
(153426) is induced by slithering the graph along itself whileinterchanging the pink and blue knots.
Erica Flapan Topological Symmetries of Molecules
A 3-connected graph
1
3
4
5 62
3-connected
(56)(23) is induced by turning the graph over.
(153426) is induced by slithering the graph along itself whileinterchanging the pink and blue knots.
(153426) is not induced by a finite order homeomorphism of S3
because there is no order 6 homeomorphism of S3 taking the figureeight knot to itself, and no knot can be the fixed point set of afinite order homeomorphism.
TSG+(Γ) = 〈(56)(23), (153426)〉 = D6. But TSG+(Γ) is notinduced by a finite group of homeomorphisms.
Erica Flapan Topological Symmetries of Molecules
A nicer embedding of Γ
Here is another embedding of Γ, where the same automorphismsare induced by finite order homeomorphisms.
Γ
1
4
52
63
Γ
1
4
3 65 2re-embed
(56)(23) is induced by turning Γ′ over left to right.
(153426) is induced by a glide rotation of Γ′ that interchanges thetwo circles while rotating counterclockwise. This glide rotation hasfinite order.
TSG+(Γ′) = D6 is induced by an isomorphic finite group of
homeomorphisms of S3.Erica Flapan Topological Symmetries of Molecules
Isometries
Finiteness Theorem [Flapan, Naimi, Pommersheim,Tamvakis]
Any 3-connected graph Γ embedded in S3 can be re-embedded asΓ′ so that TSG+(Γ) is a subgroup of TSG+(Γ
′) and is induced bya finite group of isometries of S3.
Erica Flapan Topological Symmetries of Molecules
Isometries
Finiteness Theorem [Flapan, Naimi, Pommersheim,Tamvakis]
Any 3-connected graph Γ embedded in S3 can be re-embedded asΓ′ so that TSG+(Γ) is a subgroup of TSG+(Γ
′) and is induced bya finite group of isometries of S3.
1
23
4
56
1
23
4
56
<(123)(456), (23)(56)> <(123)(456), (23)(56), (14)(25)(36)>
Γ Γ
TSG ( )=Γ +TSG ( )=Γ +
(23)(56) is induced by a finite order homeomorphism of (S3, Γ′)but not of (S3, Γ). But TSG+(Γ
′) TSG+(Γ).
Erica Flapan Topological Symmetries of Molecules
Embeddings of graphs in other manifolds
Definition
Let Γ be a graph embedded in a 3-dimensional manifold M, thenTSG(Γ,M) is the subgroup of Aut(Γ) induced byhomeomorphisms of M.
Erica Flapan Topological Symmetries of Molecules
Embeddings of graphs in other manifolds
Definition
Let Γ be a graph embedded in a 3-dimensional manifold M, thenTSG(Γ,M) is the subgroup of Aut(Γ) induced byhomeomorphisms of M.
Theorem [Flapan,Tamvakis]
Let M be a closed (i.e., compact and without boundary),connected, orientable, irreducible (i.e., can’t be split along spheresinto simpler manifolds) 3-manifold. Then there exists a finitesimple group which is not isomorphic to TSG(Γ,M) for any graphΓ embedded in M.
By our earlier result if M = S3, then G = An where n > 5.
Erica Flapan Topological Symmetries of Molecules
All finite groups are possible if M varies
Theorem [Flapan,Tamvakis]
Let M be a closed, connected, orientable, irreducible 3-manifold.Then there exists a finite simple group which is not isomorphic toTSG(Γ,M) for any graph Γ embedded in M.
The above theorem is for a fixed 3-manifold M. If we allow M tovary, then every finite group can occur.
Theorem [Flapan,Tamvakis]
For every finite group G , there is a 3-connected graph Γ embeddedin some 3-manifold M such that TSG(Γ,M) ∼= G . In fact, M canbe chosen to be a hyperbolic rational homology sphere.
Erica Flapan Topological Symmetries of Molecules
Finiteness Theorem
Recall that for S3 we had:
Finiteness Theorem [Flapan, Naimi, Pommersheim,Tamvakis]
Any 3-connected graph Γ embedded in S3 can be re-embedded asΓ′ so that TSG+(Γ) is a subgroup of TSG+(Γ
′) and is induced bya finite group of isometries of S3.
Note that S3 is Seifert fibered.
Erica Flapan Topological Symmetries of Molecules
Finiteness Theorem
Recall that for S3 we had:
Finiteness Theorem [Flapan, Naimi, Pommersheim,Tamvakis]
Any 3-connected graph Γ embedded in S3 can be re-embedded asΓ′ so that TSG+(Γ) is a subgroup of TSG+(Γ
′) and is induced bya finite group of isometries of S3.
Note that S3 is Seifert fibered.
Non-Finiteness Theorem [Flapan, Tamvakis]
For every closed, orientable, irreducible, 3-manifold M which is notSeifert fibered, there is a 3-connected graph Γ embedded in M
such that TSG+(Γ,M) is not isomorphic to any finite group ofhomeomorphisms of M.
Erica Flapan Topological Symmetries of Molecules
Thanks
H
TA
SN
K
Erica Flapan Topological Symmetries of Molecules