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The Karoubi envelope and Lee's degeneration of Khovanov homology

Dror Bar-Natan and Scott Morrison

Algebraic & Geometric Topology 6 (2006) 1459–1469

DOI: 10.2140/agt.2006.6.1459

arXiv: math.GT/0606542
Abstract

We give a simple proof of Lee's result from [Adv. Math. 179 (2005) 554--586], that the dimension of the Lee variant of the Khovanov homology of a c–component link is 2c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type theorem for tangles as well as for knots and links. Our main tool is the “Karoubi envelope of the cobordism category”, a certain enlargement of the cobordism category which is mild enough so that no information is lost yet strong enough to allow for some simplifications that are otherwise unavailable.

Keywords

categorification, cobordism, Karoubi envelope, Jones polynomial, Khovanov, knot invariants
Mathematical Subject Classification

Primary: 57M25

Secondary: 18E05, 57M27
References
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Publication

Received: 29 June 2006
Accepted: 20 July 2006
Published: 4 October 2006
Authors
Dror Bar-Natan
Department of Mathematics
University of Toronto
Toronto Ontario M5S 2E4
Canada
http://www.math.toronto.edu/~drorbn
Scott Morrison
Department of Mathematics
University of California, Berkeley
Berkeley CA 94720
USA
http://math.berkeley.edu/~scott