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


Knot Problem


Gordian Knot Problem


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


Kurt Reidemeister1931


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) )



To each knot Kthere exists an equivalen knot K'and a 2-pi periodic function f



so that( x(t), y(t), z(t) ) = ( -f(t), f '(t), -f "(t) )



Each isotopy class of knots can be described by a class of holonomic functions


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


f(t) = sin(t) + 4sin(2t) + sin(4t)


No general method was known


No general method was known

Introducing an algorithm to compute a holonomic parametrization of given knots


Some properties of holonomic knots:

Counter-clockwise orientation


Some properties of the holonomic knots:

Our Method

General observation:In AFL, not all parts are counter-clockwise

Our Method

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


Morse Knot

Projection functions

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 ) }


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 ) }



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 !

Algorithms and Data Structures for Low-Dimensional Topology

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