Dror Bar-Natan: Publications:

Khovanov's Homology for Tangles and Cobordisms

last updated Jul. 23, 2012
first finished edition: Oct. 21, 2004.
first unfinished edition: May 24, 2004.
Geometry and Topology 9-33 (2005) 1443-1499.

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a "TQFT") to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.

The paper. Cobordism.pdf, Cobordism.ps.gz, Cobordism.tar.gz.

The published version. Geometry and Topology 9-33 (2005) 1443-1499 and arXiv:math.GT/0410495.

Related handouts. Warszawa (July 2003, PDF), Harvard (October 2004, PDF), GWU (May 2004, PDF), Buffalo (October 2004, PDF).

Also see our earlier paper on the subject, "On Khovanov's Categorification of the Jones polynomial", and the papers by Khovanov, arXiv:math.QA/9908171, arXiv:math.QA/0103190 and arXiv:math.QA/0207264 and Jacobsson, arXiv:math.GT/0206303.

The cubic saddle* (z, 3Re(z3)) in the standard can [|z|<1]x[-1,1], a cup and a cap.
(see the paper, page 15)

(JavaView applet, left click and drag to rotate, right click for help and further options)

Mathematica code:

    {r Cos[t], r Sin[t], 3r^3 Cos[3t]},
    {r Cos[t], r Sin[t], 1 + r^2} / 2,
    {r Cos[t], r Sin[t], -1 - r^2} / 2
  {r, 0, 1}, {t, 0, 2Pi},
  PlotRange -> {-1, 1},
  Boxed -> False, Axes -> False,
  PlotPoints -> {10, 30},
  ViewPoint -> 0.8{2.4, -1.2, 1.5}

* The cubic saddle is also known as the "monkey saddle", as it comfortably seats a monkey with two legs and a tail.