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The Tutte polynomial is a fundamental invariant of graphs and matroids. Inthis article, we define a generalization of the Tutte polynomial to orientedgraphs and regular oriented matroids. To any regular oriented matroid $N$, weassociate 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 unorientedmatroid underlying $N$,  2. when the oriented matroid $N$ corresponds to an unoriented matroid (thatis, when the elements of the ground set come in pairs with oppositeorientations), the $A$-polynomial is equivalent to the Tutte polynomial of thisunoriented matroid (up to a change of variables),  3. the A-polynomial $A_N$ detects, among other things, whether $N$ is acyclicand whether $N$ is totally cyclic.  We explore various properties and specializations of the A-polynomial. Weshow that some of the known properties or the Tutte polynomial of matroids canbe extended to the A-polynomial of regular oriented matroids. For instance, weshow that a specialization of $A_N$ counts all the acyclic orientationsobtained by reorienting some elements of $N$, according to the number ofreoriented elements.

Tutte polynomials for regular oriented matroids