Presentation on theme: "Algorithms and Data Structures for Low-Dimensional Topology"— Presentation transcript:
1 Algorithms and Data Structures for Low-Dimensional Topology Alexander GamkrelidzeTbilisi State UniversityTbilisi,
2 Contents General ideas and remarks Description of old ideas Description of actual problemsAlgorithm to compute the holonomic parametrization of knotsAlgorithm to compute the Kontsevich integral for knotsFurther work and open problems
3 General Ideas Alles Gescheite ist schon gedacht worden, man muß nur versuchen,es noch einmal zu denkenEverything clever has beenthought already, we shouldjust try to rethink itGoethe
4 General Ideas Rethink Old Ideas in New Light !!! Application to Actual ProblemsNew Interpretation of Old Ideas
27 Holonomic Parametrization Victor Vassiliev, 1997Each 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 knotsJ. S. Birman, N. C. Wrinckle, 2000
94 Our method Advantages: The number of summands decreases Integrand functions of the same type
95 OutlookCan 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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