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

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.