Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Edge vectors on plabic networks in the disk and amalgamation of totally
  non-negative Grassmannians   Amalgamation in the totally non-negative part of positroid varieties is
equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has
proposed to represent amalgamation in positroid varieties by equivalence
classes of relations on bipartite graphs and identify total non-negativity via
edge signatures. Here we provide a complete and explicit characterization of
such signatures on the planar bicolored trivalent directed perfect networks in
the disk parametrizing positroid cells $S_M^{TNN}$.To a graph $G$ representing
$S_M^{TNN}$, we associate a geometric signature satisfying full rank condition
and total non--negativity. Such signature is uniquely identified by geometric
indices ruled by orientation and gauge ray direction. All geometric signatures
are equivalent up to a gauge transformation and complete. A system of relations
on $G$ has full rank and its image is totally non-negative for any choice of
positive weights if only if its signature is geometric; in such case, the image
is $S_M^{TNN}$. We give a combinatorial representation of geometric signatures
by proving that the total geometric signature of a face just depends on the
number of white vertices. We solve the system of geometric relations: edge
vector components are rational in the weights with subtraction--free
denominators, and have explicit expressions in terms of conservative and edge
flows. At boundary sources they give the boundary measurement matrix. If $G$ is
acyclically orientable, all components are subtraction-free rational in the
weights w.r.t. a convenient basis. We provide explicit formulas for the
transformation rules w.r.t. changes the orientation, the several gauges of the
given network, moves and reductions of networks. We express gauge freedoms and
changes of orientation as equivalence transformations of the geometric
signature.

12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex