Matroidal frameworks for topological Tutte polynomials

Abstract

We introduce the notion of a delta-matroid perspective. A delta-matroid perspective consists of a triple (M,D,N), where M and N are matroids and D is a delta-matroid such that there are strong maps from M to the upper matroid of D and from the lower matroid of D to N. We describe two Tutte-like polynomials that are naturally associated with delta-matroid perspectives and determine various properties of them. Furthermore, we show when the delta-matroid perspective is read from a graph in a surface our polynomials coincide with B. Bollobás and O. Riordan's ribbon graph polynomial and the more general Krushkal polynomial of graphs in surfaces. This is analogous to the fact that the Tutte polynomial of a graph G coincides with the Tutte polynomial of its cycle matroid. We use this new framework to prove results about the topological graph polynomials that cannot be realised in the setting of cellularly embedded graphs.