Knots-quivers correspondence

Abstract

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