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Discrete Morse Theory for Khovanov Homology

Tuomas Kelomäki

Abstract

The standard methods for calculating Khovanov homology rely either on long exact/spectral sequences or on the algorithmic "divide and conquer" approach developed by Bar-Natan. In this paper, we employ an alternative and arguably simpler tool, discrete Morse theory, which is new in the context of knot homologies. The method is applied for 2- and 3-torus braids in Bar-Natan's dotted cobordism category, where Khovanov complexes of tangles live. This grants a recursive description of the complexes of 2- and 3-torus braids yielding an inductive result on integral Khovanov homology of links containing those braids. The result, accompanied with some computer data, advances the recent progress on a conjecture by Przytycki and Sazdanović which claims that closures of 3-braids only have 2-torsion in their Khovanov homology.

Discrete Morse Theory for Khovanov Homology

Abstract

The standard methods for calculating Khovanov homology rely either on long exact/spectral sequences or on the algorithmic "divide and conquer" approach developed by Bar-Natan. In this paper, we employ an alternative and arguably simpler tool, discrete Morse theory, which is new in the context of knot homologies. The method is applied for 2- and 3-torus braids in Bar-Natan's dotted cobordism category, where Khovanov complexes of tangles live. This grants a recursive description of the complexes of 2- and 3-torus braids yielding an inductive result on integral Khovanov homology of links containing those braids. The result, accompanied with some computer data, advances the recent progress on a conjecture by Przytycki and Sazdanović which claims that closures of 3-braids only have 2-torsion in their Khovanov homology.
Paper Structure (13 sections, 23 theorems, 91 equations, 17 figures, 2 tables)

This paper contains 13 sections, 23 theorems, 91 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic Discrete Morse theory
  4. Khovanov complexes of tangles
  5. Enhanced words of a braid diagram
  6. Discrete Morse theory for 2-torus braids
  7. Adding a full twist to 2-torus braids
  8. Discrete Morse theory for 3-torus braids
  9. Adding a full twist to 3-torus braids
  10. Composing tangles and Khovanov homology
  11. Comparing homological gradings
  12. Comparing homological and internal gradings
  13. Duality and algorithmic reduction of full twists

Key Result

Theorem 1.1

Let $L:{\mathbb {Z}}^2 \to \{\text{Links}\}$ be any function which creates a link $L(k,m)$ by uniformly gluing together a fixed tangle, a 3-torus braid with $k$ twists $(\sigma_1\sigma_2)^k$ and a 2-torus braid with $m$ twists $\sigma_1^m$; see Figure Link diagram for simplified Theorem. Then, Khova

Figures (17)

  • Figure 1: A link diagram $L(k,m)$ representing an example function $L$ for Theorem \ref{['simplified algorithm theorem for introduction']}. The boxes $(\sigma_1\sigma_2)^k$ and $\sigma_1^m$ exhibit the twisting of 3 and 2 strings for $k$ and $m$ times respectively, see Figures \ref{['T2 braid figure']} and \ref{['Braid diagram of T3']} for more specific illustrations.
  • Figure 2: A braid diagram for word $\sigma_1 \sigma_2 \sigma_2^{-1} \sigma_1 \sigma_2 \sigma_1^{-1}$ (left) and its braid closure (right).
  • Figure 3: Local relations of dotted cobordisms.
  • Figure 4: Khovanov complex of $\sigma_1^{-2}$ and one of its morphisms (gradings omitted). Contrary to our usual conventions, the braid $\sigma_1^{-2}$ is rotated 90 degrees clockwise so that the cobordism $f_4$ is easier to perceive.
  • Figure 5: Sign and smoothing conventions for crossings.
  • ...and 12 more figures

Theorems & Definitions (44)

  • Theorem 1.1
  • Conjecture 1.2: Przytycki, Sazdanović
  • Lemma 2.1
  • proof
  • Theorem 2.2: Discrete Morse theory
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • ...and 34 more