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The flip map and involutions on Khovanov homology

Daren Chen, Hongjian Yang

Abstract

The flip symmetry on knot diagrams induces an involution on Khovanov homology. We prove that this involution is determined by its behavior on unlinks; in particular, it is the identity map when working over $\mathbb{F}_2$. This confirms a folklore conjecture on the triviality of the Viro flip map. As a corollary, we prove that the symmetries on the transvergent and intravergent diagrams of a strongly invertible knot induce the same involution on Khovanov homology. We also apply similar techniques to study the half sweep-around map.

The flip map and involutions on Khovanov homology

Abstract

The flip symmetry on knot diagrams induces an involution on Khovanov homology. We prove that this involution is determined by its behavior on unlinks; in particular, it is the identity map when working over . This confirms a folklore conjecture on the triviality of the Viro flip map. As a corollary, we prove that the symmetries on the transvergent and intravergent diagrams of a strongly invertible knot induce the same involution on Khovanov homology. We also apply similar techniques to study the half sweep-around map.

Paper Structure

This paper contains 15 sections, 25 theorems, 104 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The flip symmetry
  3. Algebraic and topological identifications of Khovanov chain complexes
  4. A corollary for strongly invertible knots
  5. Strategy
  6. Background
  7. Conventions for Khovanov homology
  8. Tangle invariants
  9. Involutive Khovanov homology
  10. The flip map
  11. The half sweep-around map
  12. Strongly invertible knots
  13. Discussion
  14. Comparison to involutive Heegaard Floer homology
  15. Variants of Khovanov homology

Key Result

Theorem 1.4

The flip map $\kappa\colon\mathop{\mathrm{CKh}}\nolimits(D)\to\mathop{\mathrm{CKh}}\nolimits(D)$ is chain homotopic to the identity map on $\mathop{\mathrm{CKh}}\nolimits(D)$.

Figures (15)

  • Figure 1: The flip map on the right-handed trefoil. The dashed line is the rotational axis. The labels $1^*,2^*,3^*$ are the crossings corresponding to $1,2,3$ respectively under the map $\eta$.
  • Figure 2: Left: maps between two closures of a $1$-$1$ tangle (left). Right: maps from $\widehat{\beta}$ to $\widehat{\beta^*}$.
  • Figure 3: Intravergent (left) and transvergent (right) diagrams of the right-handed trefoil. The axis is indicated by a dot in the intravergent diagram and by a dashed line in the transvergent diagram.
  • Figure 4: $0$- and $1$- smoothings
  • Figure 5: The flip map for the unknot $U$
  • ...and 10 more figures

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.4
  • Theorem 1.4
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1: khovanov2002functor*Theorem 1
  • Lemma 2.2: khovanov2002functor*Proposition 3
  • Lemma 2.3: khovanov2006invariant*Theorem 1
  • ...and 40 more