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Algorithms and Data Structures for Low-Dimensional Topology

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

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Algorithms and Data Structures for Low-Dimensional Topology

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  1. Algorithms and Data Structures for Low-Dimensional Topology Alexander Gamkrelidze Tbilisi State University Tbilisi, 7. 08. 2012

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

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

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

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

  21. 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. 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 • Naturalconnection to finite typeinvariants of knots (Vassilievinvariants) • 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 Someproperties of holonomicknots: Counter-clockwiseorientation

  33. Holonomic Parametrization Someproperties of the holonomicknots:

  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

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