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Dror Bar-Natan:
Talks:
# Local Khovanov Homology

### April 18, 2005

**Abstract. ** Many fundamental concepts in mathematics are the results
of forgetting something about even more fundamental concepts. Many binomial
identities, for example, are the *q=1* specializations of even more
powerful "*q*-binomial" identities, whose real home is in some
non-commutative universes. Likewise the Euler characteristic of a space is
the result of forgetting something about its homology; it is a beautiful and
powerful number, but the loss is noticed - homology is a much stronger
concept.

Recently it has become clear through the work of Khovanov that many knot
invariants, and perhaps many other algebraic objects, are in some sense the
Euler characteristics of objects that are potentially much more
interesting.

In my talk I will quickly review the easy and elegant Jones
polynomial of knots (a "*q*-object" in itself!). Then I will
display one complicated picture and discuss it at length. After some
definitions and some intepretation, we will see how it realizes
Khovanov's idea in a local manner, leading to some theoretical and
computational advantages.

**The picture:**

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

**Transparencies: ** Reid2Proof.pdf, R3Full.pdf, T95.html, FrameRack.pdf, KRC.pdf.

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