Deletion-Contraction and the Surface Tutte Polynomial

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, Bollobas–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.