The main result of this paper is a Pfaffian formula for the partitionfunction of the dimer model on a graph G embedded in a closed, possiblynon-orientable surface S. This formula is suitable for computational purposes,and it is obtained using purely geometrical methods. The key step in the proofconsists of a correspondence between some orientations on G and the set ofpin^- structures on S. This generalizes (and simplifies) the results of aprevious paper [2].
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