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

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 $F\ne D^2$ (except for the homology with $\mathbb{Z}∕2$ 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 $L$.