Algebraic & Geometric Topology

The Karoubi envelope and Lee's degeneration of Khovanov homology

Dror Bar-Natan and Scott Morrison

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

categorification cobordism Karoubi envelope Jones polynomial Khovanov knot invariants


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.

Export citation


  • D Bar-Natan, Fast Khovanov Homology Computations,\char'176 drorbn/papers/FastKh/
  • D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
  • P Freyd, Abelian categories. An introduction to the theory of functors, Harper's Series in Modern Mathematics, Harper & Row Publishers, New York (1964)
  • G Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996) 109–151
  • E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554–586
  • B Mazur, What is $\ldots$ a motive?, Notices Amer. Math. Soc. 51 (2004) 1214–1216
  • J A Rasmussen, Khovanov homology and the slice genus
  • S M Wehrli, A spanning tree model for Khovanov homology
  • Wikipedia, Image (category theory) –- Wikipedia, The Free Encyclopedia, } (2006) [Online; accessed 20-June-2006]
  • Wikipedia, Karoubi envelope –- Wikipedia, The Free Encyclopedia, (2006) [Online; accessed 20-June-2006]