Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 7

Knots-quivers correspondence

Pages: 1849 – 1902

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n7.a4

Authors

Piotr Kucharski (Faculty of Physics, University of Warsaw, Poland)

Markus Reineke (Faculty of Mathematics, Ruhr-Universität Bochum, Germany)

Marko Stošić (CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal; and Mathematical Institute SANU, Beograd, Serbia)

Piotr Sulkowski (Faculty of Physics, University of Warsaw, Poland; and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Cal., U.S.A.)

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.

This work was supported by the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union’s Seventh Framework Programme, and by the Foundation for Polish Science. M.S. was partially supported by the Ministry of Science of Serbia, project no. 174012.