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 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.