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A $y$-ification of Khovanov homology

Taketo Sano

Abstract

Motivated by the $y$-ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the $\mathfrak{sl}_2$-action of Gorsky, Hogancamp, and Mellit, we construct $y$-ifications of Khovanov homology and its equivariant versions within Bar-Natan's framework for tangles, and define an action of the element $e$ in $\mathfrak{sl}_2$ on these $y$-ifications. We then prove that our construction is compatible with the previous ones under Rasmussen's spectral sequence from HOMFLY--PT homology to Khovanov homology. Our construction is elementary and well suited to diagrammatic manipulations and algorithmic implementations. As a result, we verify directly that these additional structures distinguish pairs of knots with identical Khovanov homology and HOMFLY--PT homology, in particular the Conway knot and the Kinoshita--Terasaka knot.

A $y$-ification of Khovanov homology

Abstract

Motivated by the -ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the -action of Gorsky, Hogancamp, and Mellit, we construct -ifications of Khovanov homology and its equivariant versions within Bar-Natan's framework for tangles, and define an action of the element in on these -ifications. We then prove that our construction is compatible with the previous ones under Rasmussen's spectral sequence from HOMFLY--PT homology to Khovanov homology. Our construction is elementary and well suited to diagrammatic manipulations and algorithmic implementations. As a result, we verify directly that these additional structures distinguish pairs of knots with identical Khovanov homology and HOMFLY--PT homology, in particular the Conway knot and the Kinoshita--Terasaka knot.
Paper Structure (41 sections, 86 theorems, 393 equations, 16 figures)

This paper contains 41 sections, 86 theorems, 393 equations, 16 figures.

Table of Contents

  1. Introduction
  2. ${\mathcal{A}}$-modules and $y$-ifications
  3. Differential graded algebra ${\mathcal{A}}$
  4. $y$-ifications from ${\mathcal{A}}$-modules
  5. Reduction of central ${\mathcal{A}}$-modules
  6. Homotopy coherent ${\mathcal{A}}$-morphisms
  7. Transferring ${\mathcal{A}}$-actions
  8. $y$-ification of Khovanov homology
  9. Preliminaries
  10. The category of dotted cobordisms
  11. Khovanov homology for tangles
  12. Signing convention
  13. Braid diagrams
  14. Dot-sliding homotopies
  15. ${\mathcal{A}}$-action on Khovanov complex
  16. ...and 26 more sections

Key Result

Proposition 1.1

If two points $p, q$ lie on the same strand and are separated by a single crossing $c$, then there is a bidegree $(-1, 2)$ homotopy $\chi_c$ defined by the reversal of the local differential at $c$, giving

Figures (16)

  • Figure 1:
  • Figure 2: Computation of ${\mathcal{H}}(7_1)$ and $\widetilde{\mathit{Kh}}{}(7_1^*)$
  • Figure 3: Distinguishing $K_C$ and $K_{KT}$ by the action of $e$
  • Figure 4: A planar diagram
  • Figure 5: Endpoints of a crossing $c$ and an admissible coloring.
  • ...and 11 more figures

Theorems & Definitions (150)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1.3
  • Theorem 4
  • Proposition 1.4
  • Theorem 5
  • Definition 2.1
  • ...and 140 more