# Weight systems and invariants of graphs and embedded graphs

## TL;DR

This paper reviews recent advances in the theory of weight systems, focusing on their constructions from graph invariants and Lie algebra representations, including explicit formulas and recurrence relations for key cases.

## Contribution

It introduces new constructions of weight systems from graph invariants and Lie algebras, and discusses extending these concepts to embedded graphs and link invariants.

## Key findings

- Explicit generating functions for (2) weight system values
- Recurrence relations for (N) weight system computations
- Extension of weight systems to arbitrary embedded graphs

## Abstract

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).