Partial-dual genus polynomial as a weight system

TL;DR

This paper demonstrates that the partial-dual genus polynomial functions as a weight system in Vassiliev knot invariants by satisfying the four-term relation on chord diagrams.

Contribution

It establishes that the partial-dual genus polynomial is a valid weight system, linking topological graph invariants with knot theory.

Findings

Satisfies the four-term relation on chord diagrams

Validates the polynomial as a weight system in knot invariants

Bridges graph theory and knot theory concepts

Abstract

We prove that the partial-dual genus polynomial considered as a function on chord diagrams satisfies the four-term relation. Thus it is a weight system from the theory of Vassiliev knot invariants.