The transition matroid of a 4-regular graph: an introduction

TL;DR

This paper introduces a binary matroid associated with 4-regular graphs, linking it to various graph and knot polynomials through parametrized Tutte polynomial versions.

Contribution

It defines the transition matroid for 4-regular graphs and connects it to multiple well-known graph and knot polynomials.

Findings

The transition matroid encodes key properties of 4-regular graphs.

Parametrized Tutte polynomials of the matroid recover classical polynomials.

The approach unifies graph and knot polynomial theories.

Abstract

Given a 4-regular graph $F$, we introduce a binary matroid $M_{\tau}(F)$ on the set of transitions of $F$. Parametrized versions of the Tutte polynomial of $M_{\tau}(F)$ yield several well-known graph and knot polynomials, including the Martin polynomial, the homflypt polynomial, the Kauffman polynomial and the Bollob\'as-Riordan polynomial.