Semimatroids and their Tutte polynomials

TL;DR

This paper introduces semimatroids, a new class of combinatorial objects modeling affine hyperplane arrangements, and explores their Tutte polynomials, establishing their fundamental properties and interpretations.

Contribution

It defines semimatroids, characterizes their flats via geometric semilattices, and demonstrates that their Tutte polynomial is a universal invariant with combinatorial significance.

Findings

Semimatroids generalize affine hyperplane arrangements.

The Tutte polynomial of a semimatroid is universal among invariants.

Non-negative coefficients of the Tutte polynomial have a combinatorial interpretation.

Abstract

We define and study "semimatroids", a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. We show that geometric semilattices are precisely the posets of flats of semimatroids. We define and investigate the Tutte polynomial of a semimatroid. We prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial interpretation for its non-negative coefficients.