arXiv subjects
Publications and source records attributed to Maya Thompson.
The surface Tutte polynomial has recently been generalised to pseudo-surfaces equipping it with recursive deletion-contraction relations. We use these relations to show that this generalisation naturally possesses a quasi-tree expansion. This extends quasi-tree expansions of the Bollob\'as-Riordan, Las Vergnas and Krushkal polynomials, which we recover from our main result.
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. Analogous tensor product formulas are known for the ribbon graph polynomial and transition polynomials of graphs embedded in surfaces, as well as for the Bollob\'as-Riordan polynomial in some special cases. We define the tensor product of graphs embedded in pseudo-surfaces and use this to generalize and unify all of the above results, providing Brylawski-style formulas for both the Bollob\'as-Riordan and Krushkal polynomials.
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.
In this paper we unify two families of topological Tutte polynomials. The first family is that coming from the surface Tutte polynomial, a polynomial that arises in the theory of local flows and tensions. The second family arises from the canonical Tutte polynomials of Hopf algebras. Each family includes the Las Vergnas, Bollob\'as-Riordan, and Krushkal polynomials. As a consequence we determine a deletion-contraction definition of the surface Tutte polynomial and recursion relations for the number of local flows and tensions in an embedded graph.