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Quadruply-graded colored homology of knots. (English) Zbl 1432.57026
Summary: 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.

MSC:
57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
20C08 Hecke algebras and their representations
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