Volume 4, issue 2 (2004)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
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


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.

Khovanov homology, categorification, skein module, Kauffman bracket
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25, 57R56
Forward citations
Received: 23 September 2004
Revised: 6 December 2004
Accepted: 6 December 2004
Published: 15 December 2004
Marta M Asaeda
Dept of Mathematics
14 MacLean Hall
University of Iowa
Iowa City IA 52242
Jozef H Przytycki
Dept of Mathematics
Old Main Building
The George Washington University
1922 F St NW
Washington DC 20052
Adam S Sikora
Dept of Mathematics
244 Mathematics Building
SUNY at Buffalo
Buffalo NY 14260
Institute for Advanced Study
School of Mathematics
Princeton NJ 08540