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A Glimpse of the Khovanov Homology of T(2,n) Via Long Exact Sequence

Gabriel Montoya-Vega

TL;DR

This expository work develops a KH construction from the Kauffman bracket polynomial and leverages a long exact sequence to categorify the Kauffman skein relation, enabling practical KH computations for torus links $T(2,n)$ via induction. The authors derive a closed-form description of $H_{a,b}(T(2,n))$, showing KH is supported on two diagonals with explicit $\\ ext{Z}$-summands and conditional $\\text{Z}_2$-torsion depending on parity, and they provide KH tables for $T(2,11)$ and $T(2,12)$ to illustrate the pattern. The LES-based method offers a transparent, deviceful framework for calculating KH and suggests natural generalizations to other link families, potentially connecting KH torsion phenomena with chromatic graph homology. Overall, the paper aims to popularize KH, deliver concrete computational tools, and pave avenues for extending KH computations in Latin American mathematical research.

Abstract

Khovanov homology is a powerful link invariant: a categorification of the Jones polynomial that enjoys a rich and beautiful algebraic structure. This homology theory has been extensively studied and it has become an ubiquitous topic in contemporary knot theory research. In the same spirit, the Kauffman skein relation, which allows to define the Kauffman bracket polynomial up to normalization of the unknot, can be categorified by means of a long exact sequence. In an expository style, in this article we present how to build Khovanov homology from the Kauffman bracket polynomial and construct its long exact sequence. Furthermore, we present a deviceful and practical way in which this long exact sequence can be used for the computation of the Khovanov homology of torus links of the type $T(2,n)$. This article serves as a partial translation of a Spanish paper to be published on occasion of the Encuentro Internacional de Matemáticas (International Meeting of Mathematics) celebrated at the Universidad del Atlántico in Barranquilla, Colombia in November 2023. This paper offers a first look into the world of Khovanov homology by constructing it from the Kauffman bracket polynomial, as it was first done by Oleg Viro. Moreover, it gives the reader references for further studies from leading experts such as D. Bar-Natan, M. Khovanov, S. Mukherjee, J. Przytycki, and A. Shumakovitch, among others. In particular, one of the main objectives in publishing this article (and this partial translation) is to popularize research in knot theory, more specifically on Khovanov homology in Colombia, and Latin-America in general, acting as a language bridge given that most of the literature is in English.

A Glimpse of the Khovanov Homology of T(2,n) Via Long Exact Sequence

TL;DR

This expository work develops a KH construction from the Kauffman bracket polynomial and leverages a long exact sequence to categorify the Kauffman skein relation, enabling practical KH computations for torus links via induction. The authors derive a closed-form description of , showing KH is supported on two diagonals with explicit -summands and conditional -torsion depending on parity, and they provide KH tables for and to illustrate the pattern. The LES-based method offers a transparent, deviceful framework for calculating KH and suggests natural generalizations to other link families, potentially connecting KH torsion phenomena with chromatic graph homology. Overall, the paper aims to popularize KH, deliver concrete computational tools, and pave avenues for extending KH computations in Latin American mathematical research.

Abstract

Khovanov homology is a powerful link invariant: a categorification of the Jones polynomial that enjoys a rich and beautiful algebraic structure. This homology theory has been extensively studied and it has become an ubiquitous topic in contemporary knot theory research. In the same spirit, the Kauffman skein relation, which allows to define the Kauffman bracket polynomial up to normalization of the unknot, can be categorified by means of a long exact sequence. In an expository style, in this article we present how to build Khovanov homology from the Kauffman bracket polynomial and construct its long exact sequence. Furthermore, we present a deviceful and practical way in which this long exact sequence can be used for the computation of the Khovanov homology of torus links of the type . This article serves as a partial translation of a Spanish paper to be published on occasion of the Encuentro Internacional de Matemáticas (International Meeting of Mathematics) celebrated at the Universidad del Atlántico in Barranquilla, Colombia in November 2023. This paper offers a first look into the world of Khovanov homology by constructing it from the Kauffman bracket polynomial, as it was first done by Oleg Viro. Moreover, it gives the reader references for further studies from leading experts such as D. Bar-Natan, M. Khovanov, S. Mukherjee, J. Przytycki, and A. Shumakovitch, among others. In particular, one of the main objectives in publishing this article (and this partial translation) is to popularize research in knot theory, more specifically on Khovanov homology in Colombia, and Latin-America in general, acting as a language bridge given that most of the literature is in English.
Paper Structure (6 sections, 3 theorems, 25 equations, 2 figures, 3 tables)

This paper contains 6 sections, 3 theorems, 25 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Khovanov homology via Kauffman bracket polynomial
  3. Long exact sequence of Khovanov homology
  4. Khovanov Homology of Torus links $T(2,n)$
  5. Example: KH tables for T(2,11) and T(2,12)
  6. Future directions

Key Result

Theorem 2.5

Let $D$ be a link diagram. The homology groups are invariant under Reidemeister moves of second and third type. Therefore, they are invariants of unoriented framed links. Moreover, the effect of the first Reidemeister move (positive or negative) $R_{1}$, is the shift in the homology, $H_{a,b}(R_{1+}(D))=H_{a+1,b+3}(D)$ and $H_{a,b}(R_{1-}(D))=H_

Figures (2)

  • Figure 2.2: Signs of circles after fusion (left) and split (right) of components.
  • Figure 4.1: Torus knot $T(2,n)=T(2,3)$ (center), Torus knot $T(2,n-1)=T(2,2)$ (left), and the trivial knot with framing $1-n=-2$ (right).

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Definition 4.1
  • Theorem 4.2
  • proof