Authors: Gorsky, Eugene
Gukov, Sergei
Stošić, Marko 
Title: Quadruply-graded colored homology of knots
Journal: Fundamenta Mathematicae
Volume: 243
Issue: 3
First page: 301
Last page: 311
Issue Date: 1-Jan-2018
Rank: M22
ISSN: 0016-2736
DOI: 10.4064/fm30-11-2017
Abstract: 
We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau-Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work.
Keywords: BPS invariants | Colored HOMFLYPT invariants | Differentials | Knot homology | Lie superalgebras
Publisher: Instytut Matematyczny Polskiej Akademii Nauk
Project: German Science Foundation, Research Training Group 2229

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