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⊂R³ 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≠D² (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

Primary: 57M27

Secondary: 57M25, 57R56

References

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Publication

Received: 23 September 2004
Revised: 6 December 2004
Accepted: 6 December 2004
Published: 15 December 2004

Authors