192 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
Exotic subfactors of finite depth with Jones indices (5+sqrt{13})/2 and (5+sqrt{17})/2
We prove existence of subfactors of finite depth of the hyperfinite II_1
factor with indices (5+sqrt{13})/2= 4.302... and (5+sqrt{17})/2=4.561.... The
existence of the former was announced by the second named author in 1993 and
that of the latter has been conjectured since then. These are the only known
subfactors with finite depth which do not arise from classical groups, quantum
groups or rational conformal field theory.Comment: 70 pages, Latex, using epic.sty, eepic.sty, epsf.sty, here.st
Subfactors of index less than 5, part 1: the principal graph odometer
In this series of papers we show that there are exactly ten subfactors, other
than subfactors, of index between 4 and 5. Previously this
classification was known up to index . In the first paper we give
an analogue of Haagerup's initial classification of subfactors of index less
than , showing that any subfactor of index less than 5 must appear
in one of a large list of families. These families will be considered
separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete
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
Cyclotomic integers, fusion categories, and subfactors
Dimensions of objects in fusion categories are cyclotomic integers, hence
number theoretic results have implications in the study of fusion categories
and finite depth subfactors. We give two such applications. The first
application is determining a complete list of numbers in the interval (2,
76/33) which can occur as the Frobenius-Perron dimension of an object in a
fusion category. The smallest number on this list is realized in a new fusion
category which is constructed in the appendix written by V. Ostrik, while the
others are all realized by known examples. The second application proves that
in any family of graphs obtained by adding a 2-valent tree to a fixed graph,
either only finitely many graphs are principal graphs of subfactors or the
family consists of the A_n or D_n Dynkin diagrams. This result is effective,
and we apply it to several families arising in the classification of subfactors
of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri
Constructing the extended Haagerup planar algebra
We construct a new subfactor planar algebra, and as a corollary a new
subfactor, with the `extended Haagerup' principal graph pair. This completes
the classification of irreducible amenable subfactors with index in the range
, which was initiated by Haagerup in 1993. We prove that the
subfactor planar algebra with these principal graphs is unique. We give a skein
theoretic description, and a description as a subalgebra generated by a certain
element in the graph planar algebra of its principal graph. In the skein
theoretic description there is an explicit algorithm for evaluating closed
diagrams. This evaluation algorithm is unusual because intermediate steps may
increase the number of generators in a diagram.Comment: 45 pages (final version; improved introduction
Topological Quantum Field Theories and Operator Algebras
We review "quantum" invariants of closed oriented 3-dimensional manifolds
arising from operator algebras.Comment: For proceedings of "International Workshop on Quantum Field Theory
and Noncommutative Geometry", Sendai, November 200
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