Abstract
Khovanov defined graded homology groups for links and showed that their polynomial Euler characteristic is the Jones polynomial of . Khovanov’s construction does not extend in a straightforward way to links in –bundles over surfaces (except for the homology with coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in 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 determine the coefficients of in the standard basis of the skein module of Therefore, our homology groups provide a “categorification” of the Kauffman bracket skein module of . 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 to the homology groups of the mirror image of .
Citation
Marta M Asaeda. Jozef H Przytycki. Adam S Sikora. "Categorification of the Kauffman bracket skein module of $I$–bundles over surfaces." Algebr. Geom. Topol. 4 (2) 1177 - 1210, 2004. https://doi.org/10.2140/agt.2004.4.1177
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