A few weight systems arising from intersection graphs

TL;DR

This paper demonstrates that adjacency matrices of intersection graphs from chord diagrams satisfy specific relations, leading to new weight systems related to well-known knot polynomials, and extends these ideas to marked diagrams.

Contribution

It introduces a novel connection between intersection graph adjacency matrices and weight systems, including those for the Conway, HOMFLYPT, and Kauffman polynomials, via extended 2-term relations.

Findings

Intersection graph adjacency matrices satisfy 2-term relations.

Derived weight systems include those for Conway, HOMFLYPT, and Kauffman polynomials.

Extended to marked chord diagrams with new generators and relations.

Abstract

We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of {\it marked} chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial.