An activities expansion of the transition polynomial of a multimatroid
Criel Merino, Iain Moffatt, Steven Noble
TL;DR
The paper extends Tutte-polynomial ideas to multimatroids by introducing the weighted transition polynomial and developing activity-based expansions, compatible-transversal expansions, and a transversal-equivalence framework. It unifies and generalizes classical matroid results, deltamatroid translations, and ribbon-graph topological polynomials, yielding partitions of transversals into Boolean-like blocks and explicit expressions over bases. The work provides new expansions (main results: main1, maingen2, cocomp, equiv) and shows how delta-matroids, 2- and 3-matroids model embedded graphs, leading to topological transition polynomial expansions with practical applications in topological graph theory.
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.