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Positive 3-braids, Khovanov homology and Garside theory

Álvaro Del Valle Vílchez, Juan González-Meneses, Marithania Silvero

Abstract

Khovanov homology is a powerful invariant of oriented links that categorifies the Jones polynomial. Nevertheless, computing Khovanov homology of a given link remains challenging in general with current techniques. In this work we focus on links that are the closure of positive 3-braids. Starting with a classification of conjugacy classes of 3-braids arising from the Garside structure of braid groups, we compute, for any closed positive 3-braid, the first four columns (homological degree) and the three lowest rows (quantum degree) of the associated Khovanov homology table. Moreover, the number of rows and columns we can describe increases with the infimum of the positive braid (a Garside theoretical notion). We will show how to increase the infimum of a 3-braid to its maximal possible value by a conjugation, maximizing the number of cells in the Khovanov homology of its closure that can be determined, and show that this can be done in linear time.

Positive 3-braids, Khovanov homology and Garside theory

Abstract

Khovanov homology is a powerful invariant of oriented links that categorifies the Jones polynomial. Nevertheless, computing Khovanov homology of a given link remains challenging in general with current techniques. In this work we focus on links that are the closure of positive 3-braids. Starting with a classification of conjugacy classes of 3-braids arising from the Garside structure of braid groups, we compute, for any closed positive 3-braid, the first four columns (homological degree) and the three lowest rows (quantum degree) of the associated Khovanov homology table. Moreover, the number of rows and columns we can describe increases with the infimum of the positive braid (a Garside theoretical notion). We will show how to increase the infimum of a 3-braid to its maximal possible value by a conjugation, maximizing the number of cells in the Khovanov homology of its closure that can be determined, and show that this can be done in linear time.

Paper Structure

This paper contains 12 sections, 13 theorems, 37 equations, 16 figures, 33 tables.

Table of Contents

  1. Introduction
  2. Garside structure of braid groups
  3. Left normal form of $3$-braids
  4. Some highlights of Khovanov homology and semiadequate links.
  5. Khovanov homology
  6. Semiadequate links
  7. Generating alternating diagrams of rational links
  8. The $\mathbin{\mspace{1mu}\text{ }\mspace{1mu}}_{4,3}$-shape of the Khovanov homology of closed positive 3-braids
  9. Strategy for the proof of Theorem \ref{['th:main']}
  10. 3-braids with infimum greater than 0
  11. 3-braids with infimum equal to 0
  12. The $\mathbin{\mspace{1mu}\text{ }\mspace{1mu}}_{4\lfloor p/2\rfloor+4,3\lfloor p/2\rfloor+3}$-shape for closed 3-braids with infimum $p \geq 0$

Key Result

Theorem 1.1

The Khovanov homology of a closed positive $3$-braid $\beta$ is as one of the Tables tab:H(1)--table:Kh_case_C4. Moreover, $\beta$ is conjugate to a braid belonging to either $N=\{ 1, \sigma_1, \sigma_1^2, \sigma_1\sigma_2, \sigma_1^2 \sigma_2^2, \Delta \}$ (Tables tab:H(1)--tab:H(aba), respectively

Figures (16)

  • Figure 1: Artin generators and their inverses.
  • Figure 2: $A$ and $B$-smoothing of a crossing.
  • Figure 3: Standard diagram of a rational link
  • Figure 4: A description of the $\mathbf{U}$-transformation illustrating proof of Lemma \ref{['lemma:U_transformation']}.
  • Figure 5: A description of the $\mathbf{T}_i$-transformation illustrating proof of Lemma \ref{['lemma:T_transformation']}.
  • ...and 11 more figures

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 3.1
  • Proposition 3.2: Gonzalez-Meneses_Manchon_Silvero_2018
  • ...and 23 more