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

2. 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

3. 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

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

5. General Ideas: Case Study
Gordian Knot Problem

6. General Ideas: Case Study
Gordian Knot Problem

7. General Ideas: Case Study
Knot Problem

8. General Ideas: Case Study
Gordian Knot Problem

9. General Ideas: Case Study
Knot Problem

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

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

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

13. General Ideas Important low-dimensional structure: Knot
Embedding of a circle S1 into R3
A homeomorphic mapping f : S1  R3

14. General Ideas
Studying knots
Equivalent knots
Isotopic knots

15. General Ideas: Reidemeister moves

16. 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

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

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

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

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

21. AFL Representation of knots
Old idea:
AFL Representation of knots
Arkaden Arcade
Faden Thread
Lage Position

22. 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

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

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

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

26. ( 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) )

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

28. 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

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

30. Holonomic Parametrization
No general method was known

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

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

33. Holonomic Parametrization
Some properties of the holonomic knots:

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

35. Our Method

36. Our Method

37. Our Method

38. Our Method
Non-holonomic crossings

39. Our Method
Non-holonomic crossings

40. Our Method
Holonomic Trefoil

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

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

43. Using AFLs to compute the Kontsevich integral for knots

44. Using AFLs to compute the Kontsevich integral for knots
Morse Knot

45. Using AFLs to compute the Kontsevich integral for knots
Morse Knot

46. Using AFLs to compute the Kontsevich integral for knots

76. Projection functions

77. Projection functions

78. Projection functions

79. Projection functions

80. Projection functions

81. Projection functions

82. Projection functions

83. Projection functions

84. Chord diagrams

85. Chord diagrams

86. Chord diagrams

87. Chord diagrams

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

89. 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

90. The Kontsevich integral
Lk element of the generator set

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

92. 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

93. Our method
Very special functions of same type

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

95. 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?

96. Thanks !