Geometry & Topology, Vol. 9 (2005) Paper no. 33, pages 1443--1499.

Khovanov's homology for tangles and cobordisms

Dror Bar-Natan

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

AMS subject classification. Primary: 57M25. Secondary: 57M27.

E-print: arXiv:math.GT/0410495

DOI: 10.2140/gt.2005.9.1443

Submitted to GT on 3 November 2004. Paper accepted 04 July 2005. Paper published 8 August 2005.

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Dror Bar-Natan
Department of Mathematics, University of Toronto
Toronto, Ontario M5S 3G3, Canada

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