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To any trivalent plane graph embedded in the sphere, Casals and Murphyassociate a differential graded algebra (dg-algebra), in which the underlyinggraded algebra is free associative over a commutative ring. Our first result isthe construction of a generalization of the Casals--Murphy dg-algebra tonon-commutative coefficients, for which we prove various functorialityproperties not previously verified in the commutative setting. Our secondresult is to prove that rank $r$ representations of this dg-algebra, over afield $\mathbb{F}$, correspond to colorings of the faces of the graph byelements of the Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$ so thatbordering faces are transverse, up to the natural action of$\operatorname{PGL}_{2r}(\mathbb{F})$. Underlying the combinatorics, thedg-algebra is a computation of the fully non-commutative Legendrian contactdg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do notprove as such in this paper. The graph coloring problem verifies that forLegendrian 2-weaves, rank $r$ representations of the Legendrian contactdg-algebra correspond to constructible sheaves of microlocal rank $r$. This isthe first explicit such computation of the bijection between the moduli spacesof representations and sheaves for an infinite family of Legendrian surfaces.

Differential graded algebras for trivalent plane graphs and their representations