Chromatic polynomial and the $\mathfrak{so}$ weight system

TL;DR

This paper demonstrates that the leading term of the universal rf6 weight system aligns with the chromatic polynomial of the intersection graph of a chord diagram under a particular substitution, extending previous results from the rf6 to the rf6 case.

Contribution

The paper establishes a new connection between the rf6 weight system and the chromatic polynomial, generalizing prior findings from the rf6 to the rf6 algebra.

Findings

Leading term of rf6 weight system equals the chromatic polynomial under specific substitution.

Extends previous results from rf6 to rf6 algebra.

Provides a recurrence relation for rf6 weight systems.

Abstract

In a recent paper by M.Kazarian and the second author, a recurrence for the Lie algebras $\mathfrak{so}(N)$ weight systems has been suggested; the recurrence allows one to construct the universal $\mathfrak{so}$ weight system. The construction is based on an extension of the $\mathfrak{so}$ weight systems to permutations. Another recent paper, by M. Kazarian, N. Kodaneva, and the first author, shows that under the substitution $C_m=xN^{m-1}, m=1,2,\dots,$ for the Casimir elements $C_m$, 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. In the present paper, we establish a similar result for the universal $\mathfrak{so}$ weight system. That is, we show that the leading term of the universal $\mathfrak{so}$ weight system also becomes the chromatic polynomial under a specific substitution.