| Summary: | A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial. This is further extended to a more complicated graph to obtain the twisted Alexander polynomial, which involved "twisting" by a representation. The space of all representations of a given knot complement into the general linear group of a fixed size can be described by the same graph.
This work also produces a bipartite weighted signed graph to obtain the Jones polynomial for the infinite class of pretzel knots as well as for some other constructions. This is a corollary to a stronger result that calculates the activity words for the spanning trees of the Tait graph associated to a pretzel knot diagram, and this has several other applications, as well, including the Tutte polynomial and the spanning tree model of reduced Khovanov homology.
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