Geodesic
The Karoubi envelope and Lee’s degeneration of Khovanov homology
Bar-Natan, Dror1 ; Morrison, Scott2
1Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4, Canada
2Department of Mathematics, University of California, Berkeley, Berkeley CA 94720, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 3, pp. 1459-1469
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2006.6.1459
Keywords: categorification, cobordism, Karoubi envelope, Jones polynomial, Khovanov, knot invariants

Bar-Natan, Dror  1   ; Morrison, Scott  2

1 Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4, Canada
2 Department of Mathematics, University of California, Berkeley, Berkeley CA 94720, USA 
@article{10_2140_agt_2006_6_1459,
     author = {Bar-Natan, Dror and Morrison, Scott},
     title = {The {Karoubi} envelope and {Lee{\textquoteright}s} degeneration of {Khovanov} homology},
     journal = {Algebraic and Geometric Topology},
     pages = {1459--1469},
     year = {2006},
     volume = {6},
     number = {3},
     doi = {10.2140/agt.2006.6.1459},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2006.6.1459/}
}
TY  - JOUR
AU  - Bar-Natan, Dror
AU  - Morrison, Scott
TI  - The Karoubi envelope and Lee’s degeneration of Khovanov homology
JO  - Algebraic and Geometric Topology
PY  - 2006
SP  - 1459
EP  - 1469
VL  - 6
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2006.6.1459/
DO  - 10.2140/agt.2006.6.1459
ID  - 10_2140_agt_2006_6_1459
ER  - 
%0 Journal Article
%A Bar-Natan, Dror
%A Morrison, Scott
%T The Karoubi envelope and Lee’s degeneration of Khovanov homology
%J Algebraic and Geometric Topology
%D 2006
%P 1459-1469
%V 6
%N 3
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2006.6.1459/
%R 10.2140/agt.2006.6.1459
%F 10_2140_agt_2006_6_1459
Bar-Natan, Dror; Morrison, Scott. The Karoubi envelope and Lee’s degeneration of Khovanov homology. Algebraic and Geometric Topology, Tome 6 (2006) no. 3, pp. 1459-1469. doi: 10.2140/agt.2006.6.1459

[1] D Bar-Natan, Fast Khovanov Homology Computations

[2] D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443

[3] P Freyd, Abelian categories. An introduction to the theory of functors, Harper's Series in Modern Mathematics, Harper Row Publishers (1964)

[4] G Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996) 109

[5] E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554

[6] B Mazur, What is $\ldots$ a motive?, Notices Amer. Math. Soc. 51 (2004) 1214

[7] J A Rasmussen, Khovanov homology and the slice genus

[8] S M Wehrli, A spanning tree model for Khovanov homology

[9] Wikipedia, Image (category theory) — Wikipedia, The Free Encyclopedia (2006)

[10] Wikipedia, Karoubi envelope — Wikipedia, The Free Encyclopedia (2006)

Cité par Sources :