  The mutually enriching relationship between graphs and matroids has motivated
discoveries in both fields. In this paper, we exploit the similar relationship
between embedded graphs and delta-matroids. There are well-known connections
between geometric duals of plane graphs and duals of matroids. We obtain
analogous connections for various types of duality in the literature for graphs
in surfaces of higher genus and delta-matroids. Using this interplay, we
establish a rough structure theorem for delta-matroids that are twists of
matroids, we translate Petrie duality on ribbon graphs to loop complementation
on delta-matroids, and we prove that ribbon graph polynomials, such as the
Penrose polynomial, the characteristic polynomial, and the transition
polynomial, are in fact delta-matroidal. We also express the Penrose polynomial
as a sum of characteristic polynomials.
