Algebra of shares, complete bipartite graphs and $\mathfrak{sl}_2$ weight system

Abstract

A function of chord diagrams is called a weight system if it satisfies the so-called four-term relations. Vassiliev's theory describes finite-order knot invariants in terms of weight systems. In particular, there is a weight system corresponding to the coloured Jones polynomial. This weight system is described in terms of the Lie algebra $\mathfrak{sl}_2$. According to the Chmutov-Lando theorem, the value of this weight system depends only on the intersection graph of the chord diagram. Therefore, it is possible to discuss the values of this weight system at intersection graphs. We obtain formulae for the generating functions of the values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphs. Using these formulae we prove that Lando's conjecture about the degree of the polynomial that is the value of this weight system at the projection of a graph onto the subspace of primitive elements in the Hopf algebra of graphs is true for complete bipartite graphs and for a certain wider class of graphs. We introduce the algebra of shares and the $\mathfrak{sl}_2$ weight system on shares. These are the main tools for our proof. Bibliography: 14 titles.