Department of
Applied
Mathematics

On a (yet) non-existing polynomial graph invariant

Sergei Lando

HSE

Abstract

A weight system is a function on chord diagrams satisfying Vassiliev's 4-term realtions. A construction due to D.Bar-Natan and M.Kontsevich allows one to associate a weight system to any semisimple Lie algebra. The simplest nontrivial such weight system is the one associated to the Lie algebra sl(2). It comes from a knot invariant well known under the name of colored Jones polynomial. This weight system takes values in the center of the universal enveloping algebra of sl(2), which is the algebra of polynomials in a single variable (the Casimir element). Already this weight system is extremely nontrivial, and the talk will be devoted mainly to unsolved problems about it. In particular, a theorem due to S.Chmutov and the speaker asserts that this weight system leads to a partially defined polynomial graph invariant. It is interesting whether it can be extended to arbitrary graphs in a natural way. All necessary notions will be defined.