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 ID: 594737
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Slide1
Algorithms and Data Structures for Low-Dimensional Topology
Alexander GamkrelidzeTbilisi State University
Tbilisi, 7. 08. 2012Slide2
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 Slide3
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 Slide4
General Ideas
Rethink Old Ideas in New Light !!!
Application to Actual Problems
New Interpretation of Old IdeasSlide5
General Ideas: Case Study
Gordian Knot ProblemSlide6
General Ideas: Case Study
Gordian Knot ProblemSlide7
General Ideas: Case Study
Knot ProblemSlide8
General Ideas: Case Study
Gordian Knot ProblemSlide9
General Ideas: Case Study
Knot ProblemSlide10
General Ideas
Why Low-Dimentional structures? We live in 4 dimensions Generally unsolvable problems are solvable in low dimensionsSlide11
General Ideas
Why Low-Dimentional structures? We live in 4 dimensions
Robot
motion
Computer Graphics etc.Slide12
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.Slide13
General Ideas
Important low-dimensional structure:
Knot
Embedding of a circle S
1
into R
3A homeomorphic mapping f : S1 R3Slide14
General Ideas
Studying knots
Equivalent
knots
Isotopic knotsSlide15
General Ideas: Reidemeister movesSlide16
General Ideas: Reidemeister moves
Theorem (Reidemeister):
Two knots are equivalent iff they can be transformed into one another by a finite sequence of Reidemeister movesSlide17
Old idea:
AFL Representation of knotsCarl Friedrich Gauß
1877Slide18
Old idea:
AFL Representation of knotsCarl Friedrich Gauß
1877Slide19
Old idea:
AFL Representation of knotsCarl Friedrich Gauß
1877Slide20
Old idea:
AFL Representation of knotsKurt Reidemeister
1931Slide21
Old idea:
AFL Representation of knotsArkaden Arcade
Faden Thread
Lage PositionSlide22
Application
of AFL:Solving knot problem in O(n22
n/3
)
n = number of crossings
Günter Hotz, 2008
Bulletin of the Georgian National Academy of SciencesSlide23
New results:
Using AFL to computeHolonomic parametrization of knots;
Kontsevich integral for knotsSlide24
Holonomic Parametrization
Victor Vassiliev, 1997A = (
x(t), y(t), z(t)
)Slide25
Holonomic Parametrization
Victor Vassiliev, 1997
To each knot
K
there
exists an equivalen knot
K'and a 2-pi periodic function fSlide26
Holonomic Parametrization
Victor Vassiliev, 1997
so that
(
x(t), y(t), z(t)
) = ( -f(t), f '(t), -f "(t) )Slide27
Holonomic Parametrization
Victor Vassiliev, 1997
Each isotopy class of knots can be described by a class of holonomic functionsSlide28
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 knotsJ. S. Birman, N. C. Wrinckle, 2000 Slide29
Holonomic Parametrization
f(t) = sin(t) + 4sin(2t) + sin(4t)Slide30
Holonomic Parametrization
No general method was knownSlide31
Holonomic Parametrization
No general method was knownIntroducing an algorithm to compute a holonomic parametrization of given knots Slide32
Holonomic Parametrization
Some properties of holonomic knots:Counter-clockwise
orientationSlide33
Holonomic Parametrization
Some properties of the holonomic knots:Slide34
Our Method
General observation:In AFL, not all parts are counter-clockwiseSlide35
Our MethodSlide36
Our MethodSlide37
Our MethodSlide38
Our Method
Non-holonomic crossingsSlide39
Our Method
Non-holonomic crossingsSlide40
Our Method
Holonomic TrefoilSlide41
Our Method
- Describe each curve by a holonomic function;- Combine the functions to a Fourier series(using standard methods)Slide42
Our Method
Conclusion:Linear algorithm in the number of AFL crossingsSlide43
Using AFLs to compute the Kontsevich integral for knotsSlide44
Using AFLs to compute the Kontsevich integral for knots
Morse KnotSlide45
Using AFLs to compute the Kontsevich integral for knots
Morse KnotSlide46
Using AFLs to compute the Kontsevich integral for knotsSlide47Slide48Slide49Slide50Slide51Slide52Slide53Slide54Slide55Slide56Slide57Slide58Slide59Slide60Slide61Slide62Slide63Slide64Slide65Slide66Slide67Slide68Slide69Slide70Slide71Slide72Slide73Slide74Slide75Slide76
Projection functionsSlide77
Projection functionsSlide78
Projection functionsSlide79
Projection functionsSlide80
Projection functionsSlide81
Projection functionsSlide82
Projection functionsSlide83
Projection functionsSlide84
Chord diagramsSlide85
Chord diagramsSlide86
Chord diagramsSlide87
Chord diagramsSlide88
{ ( 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 ) }{ ( z3, z4 ), ( p3, p4 ) }{ ( z3, z4 ), ( p3, p1 ) }{ ( z3, z4 ), ( p2, p3 ) }{ ( z3, z4 ), ( p2, p1 ) }Chord diagramsSlide89
{ ( 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 diagramsGenerator set LD of a given chord diagram DSlide90
The Kontsevich integral
Lk element of the generator setSlide91
Our method
Embed the AFL"Moving up" in 3D means "moving up"in 2D
Mostly parallel linesSlide92
Our method
( L1 , L3 ) :
Z
1
(t)
- Z2(t) = const( L1 , L2 ) :Z3(t) - Z4(t) = 1 + t( L2 , S2 ) :
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 iSlide93
Our method
Very special functions of same typeSlide94
Our method
Advantages:
The number of summands decreases
Integrand functions of the same typeSlide95
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?Slide96
Thanks !