The universal ${\mathfrak gl}$-weight system and the chromatic polynomial

TL;DR

This paper introduces a universal ${\mathfrak gl}$-weight system that unifies ${\mathfrak gl}(N)$ weight systems and reveals a connection to the chromatic polynomial of intersection graphs, with implications for Hopf algebra homomorphisms.

Contribution

The authors construct a universal ${\mathfrak gl}$-weight system extending previous ${\mathfrak gl}(N)$ systems and establish its relation to chromatic polynomials and Hopf algebra structures.

Findings

The universal weight system's leading term in $N$ equals the chromatic polynomial of the intersection graph.

Under specific substitutions, it defines a filtered Hopf algebra homomorphism.

The paper introduces a new Hopf algebra of permutations.

Abstract

In a recent paper Zhuoke Yang, New approaches to ${\mathfrak gl}(N)$ weight system, Izvestiya Mathematics, 2023, vol. 77:6, 150--166; arXiv:2202.12225 (2022) a construction of a weight system, which unifies ${\mathfrak gl}(N)$ weight systems for $N=1,2,\dots$, has been suggested. The construction is based on an extension of the ${\mathfrak gl}(N)$ weight systems to permutations. This universal weight system takes values in the algebra of polynomials ${\mathbb C}[N;C_1,C_2,\dots]$ in infinitely many variables. We show that under the substitution $C_m=xN^{m-1}$, $m=1,2,\dots$, the leading term in $N$ of the value of the universal ${\mathfrak gl}$ weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. Moreover, we show that under the substition $C_m=p_m N^{m-1}$, $m=1,2,\dots$, the leading term in $N$ of the value of the universal ${\mathfrak gl}$ weight system determines a flitered Hopf algebra homomorphism from the rotational Hopf algebra of permutations, which we construct in the present paper, to the Hopf algebra of polynomials ${\mathbb C}[p_1,p_2,\dots]$.