Abstract
The Tutte polynomial is a fundamental invariant of graphs and matroids. In
this article, we define a generalization of the Tutte polynomial to oriented
graphs and regular oriented matroids. To any regular oriented matroid $N$, we
associate a polynomial invariant $A_N(q,y,z)$, which we call the A-polynomial.
The A-polynomial has the following interesting properties among many others:
1. a specialization of $A_N$ gives the Tutte polynomial of the unoriented
matroid underlying $N$,
2. when the oriented matroid $N$ corresponds to an unoriented matroid (that
is, when the elements of the ground set come in pairs with opposite
orientations), the $A$-polynomial is equivalent to the Tutte polynomial of this
unoriented matroid (up to a change of variables),
3. the A-polynomial $A_N$ detects, among other things, whether $N$ is acyclic
and whether $N$ is totally cyclic.
We explore various properties and specializations of the A-polynomial. We
show that some of the known properties or the Tutte polynomial of matroids can
be extended to the A-polynomial of regular oriented matroids. For instance, we
show that a specialization of $A_N$ counts all the acyclic orientations
obtained by reorienting some elements of $N$, according to the number of
reoriented elements.