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Tautological classes and symmetry in Khovanov-Rozansky homology

Eugene Gorsky, Matthew Hogancamp, Anton Mellit

Abstract

We define a new family of commuting operators $F_k$ in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that $F_2$ satisfies ``hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen.

Tautological classes and symmetry in Khovanov-Rozansky homology

Abstract

We define a new family of commuting operators in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that satisfies ``hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen.

Paper Structure

This paper contains 44 sections, 78 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. The dg algebra $\mathcal{A}$
  3. Action in link homology
  4. Relation to $y$-ification
  5. Basic objects
  6. Hard Lefshetz and symmetry
  7. Structure of the paper
  8. Homotopy theory
  9. Basic definitions
  10. Restriction/induction functors
  11. Localization
  12. Relationship to the Hochschild cohomology
  13. Relative induction/restriction
  14. Resolutions and Koszul equivalences
  15. The dg algebra $\mathcal{A}$
  16. ...and 29 more sections

Key Result

Theorem 1.2

Conjecture conj: DGR is true for all knots.

Theorems & Definitions (191)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 181 more