Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial

TL;DR

This paper extends Brylawski's tensor product formula to the transition polynomial of multimatroids, providing a unified framework that encompasses topological and graph polynomial invariants.

Contribution

It introduces a tensor product formula for multimatroid transition polynomials, generalizing Brylawski's formula to topological graph invariants.

Findings

Derived a tensor product formula for multimatroid transition polynomials

Showed Brylawski's formula as a special case of the new general result

Connected tensor product formulas to topological graph invariants

Abstract

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. We are concerned with extensions of Brylawski's tensor product formula to the Bollobas-Riordan and transition polynomials of graphs embedded in surfaces. We give a tensor product formula for the multimatroid transition polynomial and show that Brylawski's formula and its topological analogues arise as specialisations of this more general result.