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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Categorification of the Kauffman bracket skein module of $I$–bundles over surfaces

Marta M Asaeda, Jozef H Przytycki and Adam S Sikora

Algebraic & Geometric Topology 4 (2004) 1177–1210

arXiv: math.QA/0409414

Abstract

Khovanov defined graded homology groups for links L 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 FD2 (except for the homology with 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
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25, 57R56
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Publication
Received: 23 September 2004
Revised: 6 December 2004
Accepted: 6 December 2004
Published: 15 December 2004
Authors
Marta M Asaeda
Dept of Mathematics
14 MacLean Hall
University of Iowa
Iowa City IA 52242
USA
Jozef H Przytycki
Dept of Mathematics
Old Main Building
The George Washington University
1922 F St NW
Washington DC 20052
USA
Adam S Sikora
Dept of Mathematics
244 Mathematics Building
SUNY at Buffalo
Buffalo NY 14260
USA
Institute for Advanced Study
School of Mathematics
Princeton NJ 08540
USA