Algebraic and Geometric Topology 4 (2004),
paper no. 52, pages 1177-1210.
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Marta M. Asaeda, Jozef H. Przytycki and Adam S. Sikora
Abstract.
Khovanov defined graded homology groups for links L in 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 not D^2 (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
AMS subject classification.
Primary: 57M27.
Secondary: 57M25, 57R56.
DOI: 10.2140/agt.2004.4.1177
E-print: arXiv:math.QA/0409414
Submitted: 23 September 2004.
(Revised: 6 December 2004.)
Accepted: 6 December 2004.
Published: 15 December 2004.
Notes on file formats
Marta M. Asaeda, Jozef H. Przytycki and Adam S. Sikora
Dept of Mathematics, 14 MacLean Hall
University of Iowa, Iowa City,
IA 52242, USA
and
Dept of Mathematics, Old Main Bldg, The
George Washington University
1922 F St NW, Washington, DC 20052,
USA
and
Dept of Mathematics, 244 Math Bldg, SUNY at
Buffalo
Buffalo, NY 14260, USA, and
Inst for Adv Study, School
of Math, Princeton, NJ 08540, USA
Email: asaeda@math.uiowa.edu, przytyck@gwu.edu, asikora@buffalo.edu
AGT home page
