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Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant of an oriented matrix. From linear algebra we know determinant has a volumne interpretation. Is there a volume interpretation of number of perfect matchings? We also know a genus $g$ graph has permanent in terms of $4^g$ sum of determinants. Is there a geometric analog and interpretation here?

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