Algorithms and Data Structures for Low-Dimensional Topology
description
Transcript of Algorithms and Data Structures for Low-Dimensional Topology
Algorithms and Data Structures for Low-Dimensional Topology
Alexander GamkrelidzeTbilisi State University
Tbilisi, 7. 08. 2012
Contents
General ideas and remarks Description of old ideas Description of actual problems Algorithm to compute the holonomic
parametrization of knots Algorithm to compute the Kontsevich integral
for knots Further work and open problems
General Ideas
Alles Gescheite ist schon gedacht worden, man muß nur versuchen, es noch einmal zu denken
Everything clever has been thought already, we should just try to rethink it
Goethe
General Ideas
Rethink Old Ideas in New Light !!!
– Application to Actual Problems
– New Interpretation of Old Ideas
General Ideas: Case Study
Gordian Knot Problem
General Ideas: Case Study
Gordian Knot Problem
General Ideas: Case Study
Knot Problem
General Ideas: Case Study
Gordian Knot Problem
General Ideas: Case Study
Knot Problem
General Ideas
Why Low-Dimentional structures?
- We live in 4 dimensions
- Generally unsolvable problems are solvable in low dimensions
General Ideas
Why Low-Dimentional structures?
- We live in 4 dimensions
Robot motionComputer Graphics
etc.
General Ideas
Why Low-Dimentional Topology?
- Generally unsolvable problems are solvable in low dimensions
Hilbert's 10th problem
Solvability in radicals of Polynomial equat.
General Ideas
Important low-dimensional structure:
Knot
Embedding of a circle S1 into R3
A homeomorphic mapping f : S1 R3
General Ideas
Studying knots
Equivalent knots
Isotopic knots
General Ideas: Reidemeister moves
General Ideas: Reidemeister moves
Theorem (Reidemeister):
Two knots are equivalent iff they can be transformed into one another by a finite sequence of Reidemeister moves
Old idea:AFL Representation of knots
Carl Friedrich Gauß1877
Old idea:AFL Representation of knots
Carl Friedrich Gauß1877
Old idea:AFL Representation of knots
Carl Friedrich Gauß1877
Old idea:AFL Representation of knots
Kurt Reidemeister1931
Old idea:AFL Representation of knots
Arkaden ArcadeFaden ThreadLage Position
Application of AFL:
Solving knot problem in O(n22n/3)n = number of crossings
Günter Hotz, 2008 Bulletin of the Georgian National Academy of Sciences
New results:
Using AFL to compute
Holonomic parametrization of knots;
Kontsevich integral for knots
Holonomic Parametrization
Victor Vassiliev, 1997
A = ( x(t), y(t), z(t) )
Holonomic Parametrization
Victor Vassiliev, 1997
To each knot Kthere exists an equivalen knot K'and a 2-pi periodic function f
Holonomic Parametrization
Victor Vassiliev, 1997
so that( x(t), y(t), z(t) ) = ( -f(t), f '(t), -f "(t) )
Holonomic Parametrization
Victor Vassiliev, 1997
Each isotopy class of knots can be described by a class of holonomic functions
Holonomic Parametrization
1. Natural connection to finite type invariants of knots (Vassiliev invariants)
2. Two equivalent holonomic knots can be continously transformed in one another in the space of holonomic knots
J. S. Birman, N. C. Wrinckle, 2000
Holonomic Parametrization
f(t) = sin(t) + 4sin(2t) + sin(4t)
Holonomic Parametrization
No general method was known
Holonomic Parametrization
No general method was known
Introducing an algorithm to compute a holonomic parametrization of given knots
Holonomic Parametrization
Some properties of holonomic knots:
Counter-clockwise orientation
Holonomic Parametrization
Some properties of the holonomic knots:
Our Method
General observation:In AFL, not all parts are counter-clockwise
Our Method
Our Method
Our Method
Our Method
Non-holonomic crossings
Our Method
Non-holonomic crossings
Our Method
Holonomic Trefoil
Our Method
- Describe each curve by a holonomic function;- Combine the functions to a Fourier series(using standard methods)
Our Method
Conclusion:
Linear algorithm in the number of AFL crossings
Using AFLs to compute the Kontsevich integral for knots
Using AFLs to compute the Kontsevich integral for knots
Morse Knot
Using AFLs to compute the Kontsevich integral for knots
Morse Knot
Using AFLs to compute the Kontsevich integral for knots
Projection functions
Projection functions
Projection functions
Projection functions
Projection functions
Projection functions
Projection functions
Projection functions
Chord diagrams
Chord diagrams
Chord diagrams
Chord diagrams
{ ( z1, z2 ), ( p1, p3 ) } { ( z1, z2 ), ( p1, p3 ) }{ ( z1, z2 ), ( p1, p2 ) }
{ ( z1, z4 ),( p1, p4 ) }{ ( z1, z4 ),( p1, p2 ) }{ ( z1, z4 ),( p3, p4 ) }{ ( z1, z4 ),( p2, p4 ) }
{ ( z2, z3 ), ( p4, p3 ) }{ ( z2, z3 ), ( p4, p2 ) }{ ( z2, z3 ), ( p1, p3 ) }{ ( z2, z3 ), ( p1, p2 ) }
{ ( z3, z4 ), ( p3, p4 ) }{ ( z3, z4 ), ( p3, p1 ) }{ ( z3, z4 ), ( p2, p3 ) }{ ( z3, z4 ), ( p2, p1 ) }
Chord diagrams
{ ( z1, z2 ), ( p1, p3 ) } { ( z1, z2 ), ( p3, p4 ) }{ ( z1, z2 ), ( p1, p2 ) }
{ ( z1, z4 ),( p1, p4 ) }{ ( z1, z4 ),( p1, p2 ) }{ ( z1, z4 ),( p3, p4 ) }{ ( z1, z4 ),( p2, p4 ) }
{ ( z2, z3 ), ( p4, p3 ) }{ ( z2, z3 ), ( p4, p2 ) }{ ( z2, z3 ), ( p1, p3 ) }{ ( z2, z3 ), ( p1, p2 ) }
{ ( z3, z4 ), ( p3, p4 ) }{ ( z3, z4 ), ( p3, p1 ) }{ ( z3, z4 ), ( p2, p3 ) }{ ( z3, z4 ), ( p2, p1 ) }
Chord diagrams
Generator set LD of a given chord diagram D
The Kontsevich integral
Lk element of the generator set
Our method
Embed the AFL
"Moving up" in 3D means "moving up"in 2D
Mostly parallel lines
Our method
( L1 , L3 ) :
Z1(t) - Z2(t) = const
( L1 , L2 ) :
Z3(t) - Z4(t) = 1 + t
( L2 , S2 ) :
Z5(t) - Z6(t) = 1 - t + i
( P1 , S1 ) :
Z7(t) - Z8(t) = 1 + t + i
( K1 , S3 ) :
Z9(t) - Z10(t) = 2 - t i
( F1 , S4 ) :
Z11(t) - Z12(t) = 1 + t i
Our method
Very special functions of same type
Our method
Advantages:
The number of summands decreases Integrand functions of the same type
Outlook
Can we improve algorithms based on AFL restricting the domain by holonomic knots?
Besides the computation of the Kontsevich integral, can we gain more information about (determining the change of orientation?) it using the similar type of integrand functions?
Can we use AFL to improve computations in quantum groups?
Thanks !