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

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs or matroids in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the Bollobas-Riordan and transition polynomials of graphs embedded in surfaces. We show that these formulas are instances of a more general result for multimatroids by giving a tensor product formula for the multimatroid transition polynomial and showing that Brylawski's formula and its topological analogues arise from it. Along the way we provide formulas for the transition polynomial of the two-sum and star product of multimatroids.