Algorithms and Data Structures for Low-Dimensional Topology Alexander Gamkrelidze Tbilisi 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 motion Computer 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

AFL Representation of knots Old idea: AFL Representation of knots Carl Friedrich Gauß 1877

AFL Representation of knots Old idea: AFL Representation of knots Carl Friedrich Gauß 1877

AFL Representation of knots Old idea: AFL Representation of knots Carl Friedrich Gauß 1877

AFL Representation of knots Old idea: AFL Representation of knots Kurt Reidemeister 1931

AFL Representation of knots Old idea: AFL Representation of knots Arkaden Arcade Faden Thread Lage 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 K there exists an equivalen knot K' and a 2-pi periodic function f

( x(t), y(t), z(t) ) = ( -f(t), f '(t), -f "(t) ) 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 Natural connection to finite type invariants of knots (Vassiliev invariants) 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

Chord diagrams { ( z1, z2 ), ( p1, p3 ) } { ( z1, z2 ), ( p1, p2 ) }

Chord diagrams Generator set LD of a given chord diagram D { ( 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 ) } Generator set LD of a given chord diagram D

The Kontsevich integral Lk element of the generator set

Our method Embed the AFL Mostly parallel lines "Moving up" in 3D means "moving up" in 2D

Our method ( P1 , S1 ) : ( L1 , L3 ) : Z7(t) - Z8(t) = 1 + t + i Z1(t) - Z2(t) = const ( K1 , S3 ) : ( L1 , L2 ) : Z9(t) - Z10(t) = 2 - t i Z3(t) - Z4(t) = 1 + t ( F1 , S4 ) : ( L2 , S2 ) : Z11(t) - Z12(t) = 1 + t i Z5(t) - Z6(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 !