Knots-quivers correspondence

TL;DR

This paper establishes a novel correspondence between knot invariants and quiver representation theory, providing explicit quiver constructions for many knots and deriving new formulas for knot polynomials, thereby proving the LMOV conjecture in these cases.

Contribution

It introduces the knots-quivers correspondence, explicitly identifies quivers for various knots, and derives new formulas for knot invariants, proving the LMOV conjecture for these cases.

Findings

Explicit quiver representations for many knots including all up to 6 crossings

New formulas for colored HOMFLY-PT polynomials and superpolynomials

Proof of the LMOV conjecture for knots with identified quivers

Abstract

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY-PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.