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Quadruply-graded colored homology of knots

Volume 243 / 2018

Eugene Gorsky, Sergei Gukov, Marko Stošić Fundamenta Mathematicae 243 (2018), 209-299 MSC: Primary 57M27; Secondary 81T30, 20C08. DOI: 10.4064/fm30-11-2017 Published online: 3 September 2018


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.


  • Eugene GorskyDepartment of Mathematics
    UC Davis
    One Shields Avenue
    Davis, CA 95616, U.S.A.
    National Research University
    Higher School of Economics
    Moscow, Russia
  • Sergei GukovCalifornia Institute of Technology
    Pasadena, CA 91125, U.S.A.
    Max-Planck-Institut für Mathematik
    Vivatsgasse 7
    D-53111 Bonn, Germany
  • Marko StošićCAMGSD
    Departamento de Matemática
    Instituto Superior Técnico
    Av. Rovisco Pais
    1049-001 Lisbon, Portugal
    Mathematical Institute SANU
    Knez Mihailova 36
    11000 Belgrade, Serbia

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