20 research outputs found
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
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.Comment: Version 2 was obtained by merging math.QA/0403527 (now removed) with
Version 1. This version is published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-52.abs.htm
On Haagerup's list of potential principal graphs of subfactors
We show that any graph, in the sequence given by Haagerup in 1991 as that of
candidates of principal graphs of subfactors, is not realized as a principal
graph except for the smallest two. This settles the remaining case of a
previous work of the first author.Comment: 19 page