Polynomial invariants of graphs on surfaces and virtual knots

There is a natural (Euler-Poincare) duality of graphs cellularly
embedded into a surface. The faces of one graph correspond to vertices
of its dual and the edges of the graph transversally intersect the
correspondent edges of the dual graph at a single point. We generalize
this duality to a duality with respect to a subset of edges. The dual
graph might be embedded into a different (genus) surface. Moreover
this generalized duality works for embeddings to non-orientable
surfaces as well. For graphs on surfaces there is a generalization of
the Tutte polynomial called the Bollobas-Riordan polynomial. I will
explain a relation between the Bollobas-Riordan polynomials of dual
graphs and give an application of it to the knot theory. The results
are explained in arXiv:0711.3490.