  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.
