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Abstract：V. A. Vassiliev's theory of finite type knot invariants allows one to associate to such an invariant a function on chord diagrams, which are simple combinatorial objects, consisting of an oriented circle and a tuple of chords with pairwise distinct ends in it. Such functions are called "weight systems". According to a Kontsevich theorem, such a correspondence is essentially one-to-one: each weight system determines certain knot invariant.
In particular, a weight system can be associated to any semi-simple Lie algebra. However, already in the simplest nontrivial case, the one for the Lie algebra sl(2), computation of the values of the corresponding weight system is a computationally complicated task. This weight system is of great importance, however, since it corresponds to a famous knot invariant known as the colored Jones polynomial.
The last year was a period of significant progress in understanding and computing Lie algebra weight systems, both for sl(2)- and gl(N)-weight system, for arbitrary N. New recurrence relations were deduced, which allow for a lot of explicit formulas. These methods are based on an idea, due to M. Kazarian, which suggests to extend the gl(N)-weight system to permutations.
Questions concerning possible integrability properties of the Lie algebra weight systems will be formulated.
The talk is based on work of M. Kazarian, the speaker, and the students P. Zakorko, Zhuoke Yang, and P. Zinova.
