###
21/05/2004, 11:00 — 12:00 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

```
```###
Khovanov's categorification of the Jones polynomial

Following Bar-Natan's down-to-earth approach, I will explain
Khovanov's construction which associates to a knot a certain
complex of graded vector spaces. If two knots are ambient isotopic
their complexes are homotopy equivalent (grading is preserved).
Therefore the cohomology groups of the knot complex are
knot-invariants. It turns out that the Jones polynomial of a knot
equals the graded Euler characteristic of the knot cohomology.
Khovanov derived a more general polynomial from his complexes which
is a more powerful knot invariant, as has been shown by explicit
computations. References:
- Dror Bar-Natan, "On Khovanov's categorification of the Jones
polynomial", Algebraic and Geometric Topology 2 (2002) 337-370;
math.QA/0201043.
- Mikhail Khovanov, "A functor-valued invariant of tangles",
Algebr. Geom. Topol. 2 (2002) 665-741;
math.QA/0103190.
- Mikhail Khovanov, "A categorification of the Jones polynomial",
math.QA/9908171.