Weight systems and invariants of graphs and embedded graphs

Abstract

The recent progress in the theory of weight systems, which are functions on the chord diagrams satisfying the so-called 4-relations, is described. Most attention is given to methods for constructing concrete weight systems. The two main sources of the constructions discussed are invariants of the intersection graphs of chord diagrams that satisfy the 4-term relations for graphs, and metrized Lie algebras. In the simplest non-trivial case of the metrized Lie algebra $\mathfrak{sl}(2)$ the recent results on the explicit form of the generating functions of the values of a weight system on important series of chord diagrams are presented. The computations are based on the Chmutov-Varchenko recurrence relations. Another recent result presented is the construction of recurrence relations for the values of the $\mathfrak{gl}(N)$-weight system. These relations are based on Kazarian's idea of extending the $\mathfrak{gl}(N)$-weight system to arbitrary permutations. In a number of recent papers an approach to the extension of weight systems and graph invariants to arbitrary embedded graphs was proposed, which is based on an analysis of the structure of the relevant Hopf algebras. The main principles of this approach are described. Weight systems defined on embedded graphs correspond to finite-order invariants of links ('knots' with several components). Bibliography: 65 titles.