The Karoubi envelope and Lee's degeneration of Khovanov homology

Details The Karoubi envelope and Lee's degeneration of Khovanov homology The Karoubi envelope and Lee's degeneration of Khovanov homology

2006-06-21

arxiv.org/abs/math/0606542v3

Algebr. Geom. Topol. 6 (2006) 1459-1469

Mathematics

Geometric Topology

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

Scientific paper

10.2140/agt.2006.6.1459

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.

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