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Dror Bar-Natan:
Talks:
# On Khovanov's Categorification of the Jones Polynomial

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University of Illinois at Chicago

September 5, 2003

**Abstract: ** The Jones polynomial created an industry when it
was discovered in the early eighties. Yet only about four years ago
Khovanov (arXiv:math.QA/9908171)
found that it has a simple yet very intriguing generalization - that it
is the Euler characteristic of a complex whose entire homology is
invariant. And less than two years ago it was realized by Jacobsson (arXiv:math.GT/0206303)
and Khovanov (arXiv:math.QA/0207264)
that the resulting homology theory is functorial in the appropriate
sense and leads to an invariant of 2-knots in 4-space. (And it's good
to keep in mind that functoriality is the key to algebraic topology -
without it algebraic topology can do little more than classify surfaces
by their genus).

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.

**The Handout: ** NewHandout-1.pdf.

**The Picture:**