#### Volume 6, issue 3 (2006)

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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
 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