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], 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.