Algebraic and Geometric Topology 4 (2004), paper no. 52, pages 1177-1210.

Categorification of the Kauffman bracket skein module of I-bundles over surfaces

Marta M. Asaeda, Jozef H. Przytycki and Adam S. Sikora

Abstract. Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients.
After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a `categorification' of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L.

Keywords. Khovanov homology, categorification, skein module, Kauffman bracket

AMS subject classification. Primary: 57M27. Secondary: 57M25, 57R56.

DOI: 10.2140/agt.2004.4.1177

E-print: arXiv:math.QA/0409414

Submitted: 23 September 2004. (Revised: 6 December 2004.) Accepted: 6 December 2004. Published: 15 December 2004.

Notes on file formats

Marta M. Asaeda, Jozef H. Przytycki and Adam S. Sikora

Dept of Mathematics, 14 MacLean Hall
University of Iowa, Iowa City, IA 52242, USA
Dept of Mathematics, Old Main Bldg, The George Washington University
1922 F St NW, Washington, DC 20052, USA
Dept of Mathematics, 244 Math Bldg, SUNY at Buffalo
Buffalo, NY 14260, USA, and
Inst for Adv Study, School of Math, Princeton, NJ 08540, USA


AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to