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Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in thedisk count dimer configurations with prescribed boundary conditions, and theweighted version of such matrices provides a natural parametrization of thetotally non--negative part of real Grassmannians (see Refs. [54,43,44,58,7]).In this paper we provide a geometric interpretation of such variant ofKasteleyn theorem: a signature is Kasteleyn if and only if it is geometric inthe sense of Ref. [5]. We apply this geometric characterization to explicitlysolve the associated system of relations and provide a new proof that theparametrization of positroid cells induced by Kasteleyn weighted matricescoincides with that of Postnikov boundary measurement map. Finally we useKasteleyn system of relations to associate algebraic geometric data to KPmulti-soliton solutions. Indeed the KP wave function solves such system ofrelations at the nodes of the spectral curve if the dual graph of the latterrepresents the soliton data. Therefore the construction of the divisor isautomatically invariant, and finally it coincides with that in Refs. [4,6] forthe present class of graphs.

Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk