Bicolored tilings and the totally non-negative Grassmannian

Bicolored tilings are a generalization of triangulations of a surface. These tilings naturally map to a variety of combinatorial objects, namely Postnikov diagrams, plabic graphs, quivers, and positroid cells. We will first generalize the notion of edges to hyperedges to allow them to connect any number n of vertices (n > 1), and define tilings as a surface equipped with a collection of compatible hyperedges. Bicolored tilings are considered up to isotopy, and will also be subject to two equivalences that preserve some of the combinatorial properties of the tiling. We will also define a flip/mutation on the hyperedges of a tiling, which will correspond to equivalent local transformations in other combinatorial objects. We then define the Scott map and the stellar-replacement map, drawing inspiration from their definitions in [16, p.14-15] and [4, 2.1], where these maps have already been defined for triangulations and monocolored tilings. These will allow us to map bicolored tilings onto Postnikov diagrams and plabic graphs. In particular, we establish a bijection between reduced tilings and reduced Postnikov diagrams. We will dedicate a section of this paper to discuss different classes of tilings, as well as how to construct a tiling for any given permutation. Finally, we use bicolored tiling to parametrize positroid cells in the totally non-negative Grassmannian. The construction will resemble the parametrization of these cells found in [15, 12.7] and [21, 2.17], now using bicolored tilings. This will establish a bijection between the reduction-flip-equivalence classes of tilings and the positroid cells that stratify the totally non-negative Grassmannian. Degenerations of tilings will then allow us to find tilings associated to lower-dimensional positroid cells in the same Grassmannian, which also gives us a partial ordering on bicolored tilings.