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.
Asaeda, Marta M 1 ; Przytycki, Jozef H 2 ; Sikora, Adam S 3
@article{10_2140_agt_2004_4_1177,
author = {Asaeda, Marta M and Przytycki, Jozef H and Sikora, Adam S},
title = {Categorification of the {Kauffman} bracket skein module of {I{\textendash}bundles} over surfaces},
journal = {Algebraic and Geometric Topology},
pages = {1177--1210},
year = {2004},
volume = {4},
number = {2},
doi = {10.2140/agt.2004.4.1177},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.1177/}
}
TY - JOUR AU - Asaeda, Marta M AU - Przytycki, Jozef H AU - Sikora, Adam S TI - Categorification of the Kauffman bracket skein module of I–bundles over surfaces JO - Algebraic and Geometric Topology PY - 2004 SP - 1177 EP - 1210 VL - 4 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.1177/ DO - 10.2140/agt.2004.4.1177 ID - 10_2140_agt_2004_4_1177 ER -
%0 Journal Article %A Asaeda, Marta M %A Przytycki, Jozef H %A Sikora, Adam S %T Categorification of the Kauffman bracket skein module of I–bundles over surfaces %J Algebraic and Geometric Topology %D 2004 %P 1177-1210 %V 4 %N 2 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.1177/ %R 10.2140/agt.2004.4.1177 %F 10_2140_agt_2004_4_1177
Asaeda, Marta M; Przytycki, Jozef H; Sikora, Adam S. Categorification of the Kauffman bracket skein module of I–bundles over surfaces. Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1177-1210. doi: 10.2140/agt.2004.4.1177
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