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, 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.