Download presentation

Presentation is loading. Please wait.

Published byTamia Duddy Modified over 2 years ago

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 !

Similar presentations

OK

Properties of Kernels Presenter: Hongliang Fei Date: June 11, 2009.

Properties of Kernels Presenter: Hongliang Fei Date: June 11, 2009.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on civil disobedience movement gandhi Uses of water for kids ppt on batteries Ppt on regular expression examples Ppt on hard copy devices Ppt on science working models Mp ppt online Ppt on case study of samsung Ppt on different types of computer softwares list Download ppt on gi fi technology Download ppt on indus valley civilization culture