Volume 9, issue 3 (2005)

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ISSN (print): 1465-3060
Khovanov's homology for tangles and cobordisms

Dror Bar-Natan

Geometry & Topology 9 (2005) 1443–1499

arXiv: math.GT/0410495

Abstract

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2–knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a “TQFT”) to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.

Keywords
2–knots, canopoly, categorification, cobordism, Euler characteristic, Jones polynomial, Kauffman bracket, Khovanov, knot invariants, movie moves, planar algebra, skein modules, tangles, trace groups
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
References
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Publication
Received: 3 November 2004
Accepted: 4 July 2005
Published: 8 August 2005
Proposed: Vaughan Jones
Seconded: Robion Kirby, Cameron Gordon
Authors
Dror Bar-Natan
Department of Mathematics
University of Toronto
Toronto
Ontario M5S 3G3
Canada
http://www.math.toronto.edu/~drorbn/