Tutte polynomials for regular oriented matroids

TL;DR

This paper introduces the A-polynomial, a new invariant for regular oriented matroids that generalizes the Tutte polynomial, capturing properties like acyclicity and cyclicity, and extending known matroid invariants.

Contribution

The paper defines the A-polynomial for regular oriented matroids, establishing its properties, specializations, and its relation to the Tutte polynomial, thus extending matroid invariants to oriented cases.

Findings

A-polynomial specializes to the Tutte polynomial for underlying unoriented matroids.

A-polynomial detects acyclicity and total cyclicity in oriented matroids.

A specialization of the A-polynomial counts reorientations of acyclic orientations.

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.