Twists of Plücker coordinates as dimer partition functions

The homogeneous coordinate ring of the Grassmannian Grk,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plucker coordinates. We introduce a twist map on Grk,n, related to the Berenstein-Fomin-Zelevinsky-twist, and give an explicit Laurent expansion for the

twist of an arbitrary Plucker coordinate in terms of the cluster variables associated with a fixed

Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plucker coordinate. We also relate the twist map to a maximal green sequence.