Identities between dimer partition functions on different surfaces

Given a weighted graph G embedded in a non-orientable surface , one can consider the corresponding weighted graph embedded in the so-called orientation cover of . We prove identities relating twisted partition functions of the dimer model on these two graphs. When is the Mobius strip or the Klein bottle, then is the cylinder or the torus, respectively, and under some natural assumptions, these identities imply relations between the genuine dimer partition functions Z(G) and . For example, we show that if G is a locally but not globally bipartite graph embedded in the Mobius strip, then is equal to the square of Z(G). This extends results for the square lattice previously obtained by various authors.