The Karoubi envelope and Lee's degeneration of Khovanov homology

Abstract

We give a simple proof of Lee's result from [Adv. Math. 179 (2005) 554-586; arXiv:math.GT/0210213], 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.

arXiv commentThis is the version published by Algebraic & Geometric Topology on 4 October 2006

References in corpus (2)

• Khovanov's homology for tangles and cobordisms
• Spiders for rank 2 Lie algebras

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