Abstract
We construct a new polynomial invariant of maps (graphs embedded in closed surfaces, not necessarily orientable). Our invariant is tailored to contain as evaluations the number of local flows and local tensions taking non-identity values in any given finite group.
Moreover, it contains as specializations the Krushkal polynomial, the Bollobás-Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial of the under-lying graph of the map.