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

Asaeda, Marta M; Przytycki, Jozef H; Sikora, Adam S

doi:10.2140/agt.2004.4.1177

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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

DOI: 10.2140/agt.2004.4.1177

arXiv: math.QA/0409414

Abstract

Khovanov defined graded homology groups for links $L \subset ℝ^{3}$ 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 $F \neq D^{2}$ (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

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

Volume 4, issue 2 (2004)