  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.
