Plabic links, quivers, and skein relations

Abstract

We study relations between cluster algebra invariants and link invariants. First, we show that several constructions of positroid links (permutation links, Richardson links, grid diagram links, plabic graph links) give rise to isotopic links. For a subclass of permutations arising from concave curves, we also provide isotopies with the corresponding Coxeter links. Second, we associate a point count polynomial to an arbitrary locally acyclic quiver. We conjecture an equality between the top $a$-degree coefficient of the HOMFLY polynomial of a plabic graph link and the point count polynomial of its planar dual quiver. We prove this conjecture for leaf recurrent plabic graphs, which includes reduced plabic graphs and plabic fences as special cases.