Algebraic & Geometric Topology

The Karoubi envelope and Lee's degeneration of Khovanov homology

Dror Bar-Natan and Scott Morrison

Full-text: Open access

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
Received: 29 June 2006
Accepted: 20 July 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 18E05: Preadditive, additive categories

Keywords
categorification cobordism Karoubi envelope Jones polynomial Khovanov knot invariants

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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References

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