Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians

The amalgamation of cluster varieties introduced by Fock and Goncharov in [21] plays a relevant role both mathematical physical problems. In particular, the totally non-negative part positroid is explicitly described as gluing several copies small positive Grassmannians, GrTP(1,3) GrTP(2,3), has important topological implications [53] naturally appears computation amplitude scatterings N=4 SYM theory [9], [10]. Lam [44] proposed to represent equivalence classes relations on bipartite graphs identify total non-negativity via appropriate edge signatures. this paper we provide an explicit geometric characterization such signatures setting planar bicolored trivalent directed perfect networks disk [52] parametrize cells SMTNN⊂GrTNN(k,n) using systems for n-vectors. More precisely, any graph G, associate signature satisfying full rank condition non–negativity property cell. Such uniquely identified indices (local winding intersection number) ruled orientation O gauge ray direction l G. principal result enrichment Postnikov's construction associating measurements not only boundary edges or vertices, but internal well. Indeed generalize prior results Postnikov Talaska [61] providing representation solution system network (N,O,l) G weights. At aim, assign canonical basis vectors Rn at sinks define components e (finite infinite) summations over paths from given sink. have following properties: 1) They solve (N,O,l); 2) Their are rational weights with subtraction–free denominators, expressions terms conservative flows [61]. sources they coincide entries measurement matrix defined [52]. If N acyclically orientable, all respect convenient basis. Null may occur reducible orientable; 3) We formulas transformation rules gauges network, their transformations due moves reductions networks. Finally, Kasteleyn [17]. bipartite, it known that partition functions dimer configurations conditions Plucker coordinates corresponding point Grassmannian [54], [44], [60], [1], [8]. case plabic which show function multiple minor matrix. Therefore, statistical mechanical interpretation map remains open.