On a weight system conjecturally related to $\mathfrak{sl}_2$

TL;DR

This paper introduces a new family of integer-valued weight systems based on counting even cycles in intersection graphs, linking them to the $rak{sl}_2$ weight system and emphasizing their graph-dependent nature.

Contribution

It defines the series $R_k$ of weight systems, showing their dependence solely on intersection graphs and relating them to $rak{sl}_2$ weight system coefficients.

Findings

$R_k$ counts even cycles in intersection graphs.

$R_k$ depends only on the intersection graph.

$R_k$ relates to $rak{sl}_2$ weight system coefficients.

Abstract

We introduce a new series $R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system $R_k$ on a chord diagram is a signed number of cycles of even length $2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system $R_k$ on a diagram of order exactly $2k$ coincides with the coefficient of $c^k$ in the value of the $\mathfrak{sl}_2$-weight system on the projection of the diagram to primitive elements.