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.

Notes on file formats

Dror Bar-Natan
Department of Mathematics, University of Toronto
Toronto, Ontario M5S 3G3, Canada
Email: drorbn@math.toronto.edu
URL: http://www.math.toronto.edu/~drorbn

GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to http://msp.warwick.ac.uk/.