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

By M. Asaeda, Józef H. Przytycki, Adam S. Sikora. 2004

161 citations recorded by the source corpus.

Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F 6 D 2 (except for the homol- ogy 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 ofL in the standard basis of the skein module of M. Therefore, our homology groups provide a "categorifi- cation" 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. AMS Classification 57M27; 57M25, 57R56

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e205319b3e87d69c69e4de28d06470701002923

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Matrix factorizations and link homology (2004, background): “Asaeda, Przytycki and Sikora [ APS ] on categorification of invariants of links in I-bundles over surfaces, and in Audoux and Fiedler [AF], who introduced a refined Jones polynomial and its categorification, which are only invariant under braid-like isotopies.”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, methodology): “The reduced Khovanov skein complex of K A I , as defined by Asaeda, Przytycki and Sikora [1], is the associated graded object of a filtration on the Khovanov complex for K .”
On knot Floer homology in double branched covers (2007, methodology): “In the first half of this paper, we review the skein homology, first constructed by Asaeda, Przytycki and Sikora in [1], and examine its relationship to Khovanov homology.”
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology (2008, background): “More generally, categorifications of Kauffman bracket skein modules of I–bundles over surfaces [2] certainly merit further attention.”