On Khovanov's Categorification of the Jones Polynomial

Algebraic and Geometric Topology 2-16 (2002) 337-370

first posted Sep. 2001, last updated Dec. 5, 2005

In the summer of 2001 the author of this note spent a week at Harvard University visiting David Kazhdan and Dylan Thurston. Our hope for the week was to understand and improve Khovanov's seminal work on the categorification of the Jones polynomial (see arXiv:math.QA/9908171 and arXiv:math.QA/0103190). We've hardly achieved the first goal and certainly not the second; but we did convince ourselves that there is something very new and novel in Khovanov's work both on the deep conceptual level (not discussed here) and on the shallower surface level. For on the surface level Khovanov presents invariants of links which contain and generalize the Jones polynomial but whose construction is like nothing ever seen in knot theory before. Not being able to really digest it we decided to just chew some, and then provide our output as a note containing a description of his construction, complete and consistant and accompanied by computer code and examples but stripped of all philosophy and of all the linguistic gymnastics that is necessary for the philosophy but isn't necessary for the mere purpose of having a working construction. Such a note may be more accessible than the original papers. It may lead more people to read Khovanov at the source, and maybe somebody reading such a note will figure out what the Khovanov invariants really are. Read on!

The updated paper: Categorification.pdf, Categorification.ps.gz, Categorification.tar.gz (source files).
The published version: Algebraic and Geometric Topology 2-16 (2002) 337-370 and arXiv:math.GT/0201043.
The "Quick Reference Guide": QRG.ps.gz, QRG.pdf, QRG.png.
A new (July 2003) handout! NewHandout-1.pdf (April 2003 version: NewHandout.pdf).
The appendix "Khovanov's invariant for 11 crossing prime knots": KhTable11.ps.gz, KhTable11.pdf.
The appendix "Khovanov's invariant for links with up to 10 crossings": KhForLinks.ps.gz.
The appendix "Khovanov's invariant for 11 crossing links": KhForLinks11.ps.gz.
See also: Publications: Khovanov Homology Tables.
The package Categorification.m (see also CategorificationProgramExplained.html and CategorificationProgramExplained.nb).
The package Links.m.
The data file Data.m, containing the values of the rational Khovanov invariants of all prime knots with up to 11 crossings.
The data file KhForLinks.m, containing the values of the rational Khovanov invariants of all prime links with up to 11 crossings.
See also: Talks/Calgary-010824/.

Based on the computations presented in the paper, several rather strong "phenomenological" observations about Khovanov's categorification have been made. Some are in the paper, and here are some more:

A conjecture on Khovanov's invariants by Stavros Garoufalidis: "We formulate a conjectural formula for Khovanov's invariants for alternating knots in terms of the Jones polynomial and the signature of the knot."
The support of the Khovanov's invariants for alternating knots by Eun Soo Lee: "In this article, we prove the conjecture of Bar-Natan, Garoufalidis, and Khovanov's on the support of the Khovanov's invariants for alternating knots."
Patterns in knot cohomology I by Mikhail Khovanov: "Cohomology theory of links, introduced in [arXiv:math.QA/9908171], is combinatorial. Dror Bar-Natan recently wrote a program that found ranks of cohomology groups of all prime knots with up to 11 crossings [this page]. His surprising experimental data is discussed in this note."

Finally, there is a newer and more powerful program to compute Khovanov homologies, by Alexander Shumakovitch. It's here.

(free sample - more inside!)