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Algorithms and Data Structures for Low-Dimensional Topology Alexander Gamkrelidze Tbilisi State University Tbilisi,

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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

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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

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General Ideas Rethink Old Ideas in New Light !!! – Application to Actual Problems – New Interpretation of Old Ideas

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General Ideas: Case Study Gordian Knot Problem

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General Ideas: Case Study Gordian Knot Problem

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General Ideas: Case Study Knot Problem

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General Ideas: Case Study Gordian Knot Problem

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General Ideas: Case Study Knot Problem

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General Ideas Why Low-Dimentional structures? - We live in 4 dimensions - Generally unsolvable problems are solvable in low dimensions

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General Ideas Why Low-Dimentional structures? - We live in 4 dimensions Robot motion Computer Graphics etc.

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General Ideas Why Low-Dimentional Topology? - Generally unsolvable problems are solvable in low dimensions Hilbert's 10 th problem Solvability in radicals of Polynomial equat.

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General Ideas Important low-dimensional structure: Knot Embedding of a circle S 1 into R 3 A homeomorphic mapping f : S 1 R 3

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General Ideas Studying knots Equivalent knots Isotopic knots

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General Ideas: Reidemeister moves

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Theorem (Reidemeister): Two knots are equivalent iff they can be transformed into one another by a finite sequence of Reidemeister moves

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Old idea: AFL Representation of knots Carl Friedrich Gauß 1877

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Old idea: AFL Representation of knots Carl Friedrich Gauß 1877

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Old idea: AFL Representation of knots Carl Friedrich Gauß 1877

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Old idea: AFL Representation of knots Kurt Reidemeister 1931

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Old idea: AFL Representation of knots ArkadenArcade FadenThread LagePosition

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Application of AFL: Solving knot problem in O(n 2 2 n/3 ) n = number of crossings Günter Hotz, 2008 Bulletin of the Georgian National Academy of Sciences

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New results: Using AFL to compute Holonomic parametrization of knots; Kontsevich integral for knots

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Holonomic Parametrization Victor Vassiliev, 1997 A = ( x(t), y(t), z(t) )

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Holonomic Parametrization Victor Vassiliev, 1997 To each knot K there exists an equivalen knot K' and a 2-pi periodic function f

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Holonomic Parametrization Victor Vassiliev, 1997 so that ( x(t), y(t), z(t) ) = ( -f(t), f '(t), -f "(t) )

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Holonomic Parametrization Victor Vassiliev, 1997 Each isotopy class of knots can be described by a class of holonomic functions

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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

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

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Holonomic Parametrization No general method was known

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Holonomic Parametrization No general method was known Introducing an algorithm to compute a holonomic parametrization of given knots

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Holonomic Parametrization Some properties of holonomic knots: Counter-clockwise orientation

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Holonomic Parametrization Some properties of the holonomic knots:

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

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Our Method

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Non-holonomic crossings

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Our Method Non-holonomic crossings

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Our Method Holonomic Trefoil

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Our Method - Describe each curve by a holonomic function; - Combine the functions to a Fourier series (using standard methods)

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Our Method Conclusion: Linear algorithm in the number of AFL crossings

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Using AFLs to compute the Kontsevich integral for knots

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Morse Knot

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Using AFLs to compute the Kontsevich integral for knots Morse Knot

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Using AFLs to compute the Kontsevich integral for knots

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Projection functions

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Chord diagrams

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{ ( 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

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{ ( 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 L D of a given chord diagram D

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The Kontsevich integral L k element of the generator set

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

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Our method ( L 1, L 3 ) : Z 1 (t) - Z 2 (t) = const ( L 1, L 2 ) : Z 3 (t) - Z 4 (t) = 1 + t ( L 2, S 2 ) : Z 5 (t) - Z 6 (t) = 1 - t + i ( P 1, S 1 ) : Z 7 (t) - Z 8 (t) = 1 + t + i ( K 1, S 3 ) : Z 9 (t) - Z 10 (t) = 2 - t i ( F 1, S 4 ) : Z 11 (t) - Z 12 (t) = 1 + t i

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Our method Very special functions of same type

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Our method Advantages: The number of summands decreases Integrand functions of the same type

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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?

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Thanks !

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