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Dror Bar-Natan:
Talks:
# Khovanov Homology for Tangles and Cobordisms

## Colloquium, University of Toronto

### SS5017, 4PM, December 8, 2004

**Abstract. ** In my talk I will display one complicated picture
and discuss it at length, finding that it's actually quite simple.
Applying a certain 2D TQFT, we will get a homology theory whose Euler
characteristic is the Jones polynomial. Not applying it, very cheaply
we will get an invariant of tangles which is functorial under
cobordisms and an invariant of 2-knots.

Why is it interesting?

- It has several generalizations, but as a whole, we hardly understand
it. It may have significant algebraic and/or physical ramifications. In
fact, it suggests that much of algebra as we know it (or at least quantum
algebra as we know it), is a shadow of some "higher algebra".
- It is a knot/link/tangle invariant stronger than the Jones polynomial,
and an invariant of 2-knots in 4-space.
- It seems stronger than the original "Khovanov Homology", which in
itself is stronger than the Jones polynomial.
- It is functorial in the appropriate sense, and Rasmussen (math.GT/0402131)
uses it to do some real topology.

**Prerequisites. ** Not to be scared of the words "category" and
"functor" and the phrase "a homotopy between chain complexes".
Otherwise all will be explained.

**The picture:**

**Handouts: **
MoreFormulas.pdf.
NewHandout-1.pdf,

**Transparencies: **
Reid2Proof.pdf,
R3Full.pdf,
FrameRack.pdf.

See also my paper Khovanov's
Homology for Tangles and Cobordisms.