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Comptes Rendus Mathématique
Volume 353, n° 12
pages 1135-1140 (décembre 2015)
Doi : 10.1016/j.crma.2015.09.026
Received : 17 October 2014 ;  accepted : 28 September 2015
Une généralisation de l'homologie de Khovanov
A generalization of Khovanov's homology
 

Zouhaier Aouani a, b , Inès Saihi b, c
a Université Paris-7, France 
b Université de Tunis El Manar, Faculté des sciences de Tunis, LR11ES12, Tunisie 
c Université de Tunis, École nationale supérieure d'ingénieurs de Tunis, 5, avenue Taha-Hussein, 1008, Tunisie 

Résumé

Dans ce travail, on généralise l'homologie de Khovanov à tout entrelacs non orienté et à coefficients dans l'algèbre de Frobenius universelle de rang 2, puis on construit pour chaque mouvement de Reidemeister une équivalence d'homotopie graduée, en donnant les formules explicites de ces équivalences ainsi que celles de leurs inverses homotopiques. Ensuite, on montre que le complexe de Khovanov généralisé de l'image miroir d'un diagramme d'entrelacs D est isomorphe au dual du complexe de Khovanov de D . Enfin, on généralise le théorème de Rasmussen sur l'homologie de Lee–Khovanov.

The full text of this article is available in PDF format.
Abstract

We generalize Khovanov's homology to unoriented links and with coefficients in the 2-dimensional universal Frobenius algebra. Then, we construct, for each Reidemeister move, a graduated homotopy equivalence and give the explicit formulae of these equivalences and their homotopy inverses. Further, we prove that the generalized Khovanov complex of a link diagram D mirror image is isomorphic to the dual of the Khovanov complex of D . Finally, we generalize the Rasmussen theorem on the Lee–Khovanov homology.

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