Weight systems and invariants of graphs and embedded graphs

We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these constructions that are discussed in the paper are invariants of intersection graphs of chord diagrams that satisfy $4$-term relations for graphs and metrized Lie algebras. For the simplest nontrivial metrized Lie algebra $\mathfrak {sl}(2)$, we present recent results about the explicit form of generating functions for the values of the corresponding weight system on important families of chord diagrams. We also explain another recent result: construction of recurrence relations for computing the values of the $\mathfrak{gl}(N)$-weight system. These relations are based on M. Kazarian's extension of the $\mathfrak{gl}(N)$-weight system to arbitrary permutations. Certain recent papers suggest an approach to extending weight systems and graph invariants to arbitrary embedded graphs, which is based on the study of the corresponding Hopf algebra structures; we describe this approach. Weight systems defined on arbitrary embedded graphs correspond to finite type invariants of links (multicomponent knots).