Khovanov’s homology for tangles and cobordisms

Bar-Natan, Dror

doi:10.2140/gt.2005.9.1443

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Khovanov's homology for tangles and cobordisms

Dror Bar-Natan

Geometry & Topology 9 (2005) 1443–1499

DOI: 10.2140/gt.2005.9.1443

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

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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/

Volume 9, issue 3 (2005)