Web bases in degree two from hourglass plabic graphs

Webs give a diagrammatic calculus for spaces of Uq(slᵣ)-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such Uq(sl₄)-web bases. Separately, Fraser introduced a web basis for Pl\"{u}cker degree two representations of arbitrary Uq(slᵣ). Here, we show that Fraser's basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant Uq(slᵣ)-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general Uq(slᵣ)-web bases.