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.

On the primitive subspace of Lando framed graph bialgebra

“…For instance, it is known that the values of the chromatic polynomial (that is, the numbers of 𝑘-colorings of the vertices of a graph) are 4-invariants of graphs [3,7]. On the other hand, the number of 3-colorings of a graph may be used to produce a new 4-invariant as follows.…”

“…This formula is an alternative version of the projection formula [8,13]. The reader is invited to compare this statement with the Remark to section 2.2 of [7] -it describes, implicitly, the result of applying 𝜓 to a primitive element of 𝐺 𝒞 /𝐼 𝒞 .…”

New approaches to $\mathfrak{gl}_N$ weight system

“…After the present paper has been submitted, the paper [11] appeared, where the constructions of the present paper are included in the general context of weight systems and graph invariants. In [12], we extend the results of the present paper to Lie superalgebras gl(m|n).…”