Algebraic & Geometric Topology

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 $2c$, 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.

Article information

Source
Algebr. Geom. Topol., Volume 6, Number 3 (2006), 1459-1469.

Dates
Accepted: 20 July 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796586

Digital Object Identifier
doi:10.2140/agt.2006.6.1459

Mathematical Reviews number (MathSciNet)
MR2253455

Zentralblatt MATH identifier
1130.57012

Citation

Bar-Natan, Dror; Morrison, Scott. The Karoubi envelope and Lee's degeneration of Khovanov homology. Algebr. Geom. Topol. 6 (2006), no. 3, 1459--1469. doi:10.2140/agt.2006.6.1459. https://projecteuclid.org/euclid.agt/1513796586

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