Plabic links, quivers, and skein relations

TL;DR

This paper explores the connections between cluster algebra invariants and link invariants, establishing link isotopies for various constructions and proposing a conjecture relating HOMFLY polynomial coefficients to quiver point counts, with partial proofs.

Contribution

It demonstrates that different constructions of positroid links produce isotopic links and introduces a conjecture linking HOMFLY polynomial coefficients to quiver point counts, proved for certain classes.

Findings

Positroid link constructions yield isotopic links.

Conjecture relating HOMFLY polynomial and quiver point counts proposed.

Proven for leaf recurrent plabic graphs, including reduced plabic graphs.

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.