An activities expansion of the transition polynomial of a multimatroid

TL;DR

This paper introduces an activities expansion for the weighted transition polynomial of multimatroids, generalizing Tutte polynomial results and applying to delta-matroids and embedded graphs.

Contribution

It develops a new activities expansion for the multimatroid transition polynomial, extending Tutte polynomial theories and applying to topological and delta-matroid contexts.

Findings

Derived an activities expansion for the multimatroid transition polynomial.

Decomposed transversals into boolean lattice structures.

Extended results to delta-matroids and embedded graphs.

Abstract

The weighted transition polynomial of a multimatroid is a generalization of the Tutte polynomial. By defining the activity of a skew class with respect to a basis in a multimatroid, we obtain an activities expansion for the weighted transition polynomial. We also decompose the set of all transversals of a multimatroid as a union of subsets of transversals. Each term in the decomposition has the structure of a boolean lattice, and each transversal belongs to a number of terms depending only on the sizes of some of its skew classes. Further expressions for the transition polynomial of a multimatroid are obtained via an equivalence relation on its bases and by extending Kochol's theory of compatible sets. We apply our multimatroid results to obtain a result of Morse about the transition polynomial of a delta-matroid and get a partition of the boolean lattice of subsets of elements of a delta-matroid determined by the feasible sets. Finally, we describe how multimatroids arise from graphs embedded in surfaces and apply our results to obtain an activities expansion for the topological transition polynomial. Our work extends results for the Tutte polynomial of a matroid.