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Notes on Khovanov homology

Melissa Zhang

TL;DR

These notes provide an expository tour of Khovanov homology and its surrounding framework, starting from the Jones polynomial and proceeding through Bar-Natan's cobordism-based TQFT to the construction of the bi-graded chain complex $\mathrm{CKh}(D)$ whose homology categorifies $\hat{J}$. The exposition emphasizes computational tools, functoriality, and a spectrum of applications, including Lee homology and Rasmussen's $s$-invariant, which yield strong 4‑dimensional and concordance information via filtrations and cobordism maps. The material highlights how categorification via $\mathrm{Kh}$ provides deeper invariants for links, surfaces in $B^4$, and Legendrian/transverse knot theory, with connections to spectral sequences and 4‑manifold topology. Overall, the notes trace a coherent path from skein-based invariants to powerful homological tools with rich geometric and physical interpretations, illustrating both foundational constructions and cutting-edge applications.

Abstract

These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. Major topics include the Jones polynomial, Khovanov homology, Bar-Natan's cobordism category, applications of Khovanov homology, some spectral sequences, Khovanov stable homotopy type, and skein lasagna modules. Topological and algebraic exposition are sprinkled throughout as needed.

Notes on Khovanov homology

TL;DR

These notes provide an expository tour of Khovanov homology and its surrounding framework, starting from the Jones polynomial and proceeding through Bar-Natan's cobordism-based TQFT to the construction of the bi-graded chain complex whose homology categorifies . The exposition emphasizes computational tools, functoriality, and a spectrum of applications, including Lee homology and Rasmussen's -invariant, which yield strong 4‑dimensional and concordance information via filtrations and cobordism maps. The material highlights how categorification via provides deeper invariants for links, surfaces in , and Legendrian/transverse knot theory, with connections to spectral sequences and 4‑manifold topology. Overall, the notes trace a coherent path from skein-based invariants to powerful homological tools with rich geometric and physical interpretations, illustrating both foundational constructions and cutting-edge applications.

Abstract

These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. Major topics include the Jones polynomial, Khovanov homology, Bar-Natan's cobordism category, applications of Khovanov homology, some spectral sequences, Khovanov stable homotopy type, and skein lasagna modules. Topological and algebraic exposition are sprinkled throughout as needed.
Paper Structure (86 sections, 40 theorems, 173 equations, 1 algorithm)

This paper contains 86 sections, 40 theorems, 173 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Evolution
  3. Knots and Topology
  4. Knots and links
  5. Link diagrams and Reidemeister moves
  6. Jones polynomial via skein relation
  7. Jones polynomial from Kauffman bracket
  8. Jones polynomial via the Khovanov bracket
  9. Khovanov homology
  10. The Khovanov chain complex $\mathrm{CKh}(D)$
  11. Cube of resolutions
  12. Chain groups, distinguished generators, gradings
  13. Differentials
  14. Global grading shifts and homology
  15. Algebraic aside: Mapping cones
  16. ...and 71 more sections

Key Result

Theorem 2.2.1

If $D$ and $D'$ are two diagrams of the same link, then they are related by a finite sequence of the following moves:

Theorems & Definitions (250)

  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Example 2.1.4
  • Definition 2.1.5
  • Definition 2.1.6
  • Remark 2.1.7
  • Remark 2.1.10
  • Theorem 2.2.1: Reidemeister, 1930s
  • Remark 2.2.2
  • ...and 240 more