  The homogeneous coordinate ring of the Grassmannian Gr(k,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 Pluecker
coordinates. We introduce a twist map on Gr(k,n), related to the
Berenstein-Fomin-Zelevinsky-twist, and give an explicit Laurent expansion for
the twist of an arbitrary Pluecker 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
Pluecker coordinate. We also relate the twist map to a maximal green sequence.
