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Dror Bar-Natan:
Talks:
# Talks at Michigan State University

### February 17 and 18, 2004

## Khovanov Homology for Knots and Links.

** Colloquium, February 17. **

**Abstract. **Over the last 20 years, knot theorists have been
extremely good at borrowing ideas from other fields. We've borrowed
from Mathematical Physics and borrowed from Algebra and we have a
Beautiful Theory of Knot Invariants that can claim deep heritage on
either side. But we haven't been so good at returning. While not
entirely impossible, is remains difficult to point at developments in
quantum field theory or quantum algebra (our lenders) that owe
something to our Beautiful Theory of Knot Invariants.

Came Khovanov in 1999 and changed the picture dramatically by
offering Mathematical Physics and Algebra the most valuable prize known
to mathematicians - a *challenge*. For none of them can yet
explain whence comes his "Categorification of the Jones Polynomial" - a
far reaching generalization of the most celebrated member of our
Beautiful Theory of Knot Invariants. The Mathematical Physics and
Algebra underlying the Jones polynomial are deep and substantial, and
there are all reasons to believe that a successful resolution of
Khovanov's challenge will be the same.

In my talk I will quickly describe the Jones polynomial (it's so
easy) and then move on to describe Khovanov's homological
generalization thereof.

(Khovanov's homology is also a stronger invariant than the Jones
polynomial and it is "functorial" in some 4-dimensional sense).

Handout: QRG.pdf.

Slide: Kauffman's
Bracket.

## Khovanov Homology for Tangles and Cobordisms.

** Seminar, February 18. **

**Abstract. **In my talk I will display one complicated picture and
discuss it at length. 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.

Handout side 1:
NewHandout-1.pdf.

Handout side 2:
handout2.pdf.

**The Picture:**