[Paper Review] Five Lectures on Khovanov Homology

Paul Turner|ArXiv.org|Jun 19, 2006

Geometric and Algebraic Topology32 references19 citations

TL;DR

This paper provides a comprehensive, pedagogical introduction to Khovanov homology, a categorification of the Jones polynomial, using a combinatorial construction of a bigraded chain complex from link diagrams. It establishes that Khovanov homology is a link invariant with graded Euler characteristic equal to the unnormalized Jones polynomial, and explores its functoriality, torsion, and connections to other homology theories and quantum invariants.

ABSTRACT

These lecture notes, which were designed for the Summer School "Heegaard-Floer Homology and Khovanov Homology" in Marseilles, 29th May - 2nd June, 2006, provide an elementary introduction to Khovanov homology. The intended audience is graduate students with some minimal background in low-dimensional and algebraic topology. The lectures cover the basic definitions, important properties, a number of variants and some applications. At the end of each lecture the reader is referred to the relevant literature for further reading.

Motivation & Objective

• To provide a self-contained, accessible introduction to Khovanov homology for graduate students with minimal background in low-dimensional topology.
• To explain how Khovanov homology categorifies the Jones polynomial by replacing a polynomial invariant with a bigraded homology theory.
• To explore the functorial properties, invariance under Reidemeister moves, and the role of torsion in Khovanov homology.
• To introduce variants such as Lee theory, Rasmussen's invariant, and Khovanov-Rozansky homology, and their connections to other invariants.
• To motivate and survey the broader context of link homology, including relations to graph homology, HOMFLYPT polynomials, and geometric and physical interpretations.

Proposed method

• Construct a bigraded chain complex $ C^{*,*}(D) $ from a link diagram $ D $ using smoothings of crossings (0- and 1-smoothings) and a Frobenius algebra structure.
• Define the differential on the complex using cobordism maps between smoothings, ensuring Reidemeister invariance via the TQFT framework.
• Apply homology to the complex to obtain the Khovanov homology $ KH^{*,*}(D) $, which is invariant under Reidemeister moves.
• Use the graded Euler characteristic formula $ \sum (-1)^i q^j \dim KH^{i,j}(D) = \hat{J}(D) $ to show categorification of the unnormalized Jones polynomial.
• Introduce variants such as Lee theory (with differential $ d_{\text{Lee}} $) and Rasmussen's $ s $-invariant via spectral sequences.
• Extend the framework to graph homology and Khovanov-Rozansky homology using Frobenius algebras and categorified quantum groups.

Experimental results

Research questions

• RQ1 How can the Jones polynomial be categorified into a bigraded homology theory that retains more topological information?
• RQ2 What are the functorial and topological properties of Khovanov homology, particularly its invariance under Reidemeister moves?
• RQ3 How does torsion in Khovanov homology reveal information beyond the Jones polynomial?
• RQ4 What is the relationship between Khovanov homology and other link invariants such as the HOMFLYPT polynomial and the colored Jones polynomial?
• RQ5 What geometric or physical structures underlie Khovanov homology, and how do they relate to Floer homology or quantum field theory?

Key findings

• Khovanov homology $ KH^{*,*}(D) $ is a bigraded homology theory that is invariant under Reidemeister moves, making it a link invariant.
• The graded Euler characteristic of Khovanov homology recovers the unnormalized Jones polynomial $ \hat{J}(D) $, demonstrating categorification.
• Khovanov homology detects torsion, which is not visible in the Jones polynomial, making it a stronger invariant than the polynomial alone.
• Rasmussen's $ s $-invariant, derived from a spectral sequence in Lee theory, provides a powerful concordance invariant that detects the slice genus.
• Khovanov-Rozansky homology generalizes the construction to higher $ N $, categorifying the HOMFLYPT polynomial and linking to quantum groups.
• Graph homology, constructed via a similar Frobenius algebra framework, categorifies the chromatic and dichromatic polynomials, with a long exact sequence mirroring deletion-contraction relations.