Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás–Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás–Riordan polynomials.