Triangular-grid billiards and plabic graphs

Given a polygon P in the triangular grid, we obtain a permutation πP via a natural billiards system in which beams of light bounce around inside of P. The different cycles in πP correspond to the different trajectories of light beams. We prove that area(P) ⩾ 6 cyc(P) − 6 and perim(P) ⩾⁷/₂ cyc(P) –³/₂, where area(P) and perim(P) are the (appropriately normalized) area and perimeter of P, respectively, and cyc(P) is the number of cycles in πP. The inequality concerning area(P) is tight, and we characterize the polygons P satisfying area(P) = 6 cyc(P) − 6. These results can be reformulated in the language of Postnikov’s plabic graphs as follows. Let G be a connected reduced plabic graph with essential dimension 2. Suppose G has n marked boundary points and v (internal) vertices, and let c be the number of cycles in the trip permutation of G. Then we have v ⩾ 6c − 6 and n ⩾⁷/₂c –³/₂.