The Karoubi envelope and Lee’s degeneration of Khovanov homology

Bar-Natan, Dror; Morrison, Scott

doi:10.2140/agt.2006.6.1459

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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 $2^{c}$ , 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 2000

Primary: 57M25

Secondary: 57M27, 18E05

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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

Volume 6, issue 3 (2006)