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The standard parametrization of totally non-negative Grassmannians wasobtained by A. Postnikov [45] introducing the boundary measurement map in termsof discrete path integration on planar bicolored (plabic) graphs in the disk.An alternative parametrization was proposed by T. Lam [38] introducing systemsof relations on vectors on such graphs, depending on some signatures defined onedges. The problem of characterizing the signatures corresponding to thetotally non-negative cells, was left open in [38]. In our paper we provide anexplicit construction of such signatures, satisfying both the full rankcondition and the total non-negativity property on the full positroid cell. Ifthe graph $\mathcal G$ satisfies the following natural constraint: each edgebelongs to some oriented path from the boundary to the boundary, then suchsignature is unique up to a vertex gauge transformation. Such signature isuniquely identified by geometric indices (local winding and intersectionnumber) ruled by the orientation $\mathcal O$ and gauge ray direction$\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorialrepresentation of geometric signatures by showing that the total signature ofevery finite face just depends on the number of white vertices on it. Thelatter characterization is a Kasteleyn-type property [7,1] and we conjecture amechanical-statistical interpretation of such relations. An explicit connectionbetween the solution of Lam system of relations and the value of Postnikovboundary measurement map is established using the generalization of Talaskaformula [50] obtained in [6]. In particular, the components of the edge vectorsare rational in the edge weights with subtraction-free denominators. Finally,we provide explicit formulas for transformations of signatures under Postnikovmoves and reductions, and amalgamations of networks.

Geometric nature of relations on plabic graphs and totally non-negative Grassmannians