Publications
Here are my papers.
Click on a title to download a paper in postscript format.
If you prefer some other format,
you can find most of these papers on the
arxiv.
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Skein theory for the ADE planar algebras,
(preprint).
Gives generators and relations for the ADE planar algebras.
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Generalized Long-Moody representations of braid groups,
with
Jianjun Tian
Communications in Contemporary Mathematics (2008)
Algorithms to build new representations from old.
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A homological definition of the HOMFLY polynomial
Algebraic and Geometric Topology 7 (2007)
This does for the HOMFLY polynomial
what "A homological definition of the Jones polynomial"
did for the Jones polynomial.
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Braid groups and the Iwahori-Hecke algebras
Problems on mapping class groups and related topics (2006)
Mostly a survey paper,
including why the HOMFLY polynomial is
the unique Markov trace of the Iwahori-Hecke algebra,
and open problems.
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Homological representations of the Iwahori-Hecke algebra
Geometry and Topology Monographs, 7 (2004)
Gives a construction of representations of the Iwahori-Hecke algebra
from a natural action of the braid group
on homology with local coefficients of a configuration space.
This works for non-generic values of the parameter,
and conjecturally gives all irreps.
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The Lawrence-Krammer representation
Topology and geometry of manifolds, Proc. Sympos. Pure Math., 71 (2003)
We study the action of the braid group
on the second homology of a certain configuration space
by describing surfaces representing elements of homology.
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Representations of braid groups
Proceedings of the International Congress of Mathematicians, Vol. II
(Beijing, 2002)
I don't have a copy, but here are
the slides from my ICM talk.
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A homological definition of the Jones polynomial
Geometry and Topology Monographs, vol. 4 (2002)
Obtains the Jones polynomial as an algebraic intersection pairing
between certain elements of singular homology of a configuration space.
This has nothing to do with Khovanov homology (yet).
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Does the Jones polynomial detect the unknot?
Journal of Knot Theory and its Ramifications, 2002
An algorithm to search for a non-trivial braid in
the kernel of the Burau representation of the braid group B_4.
Such a braid would almost certainly
lead to a non-trivial knot whose Jones polynomial
is equal to one.
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The mapping class group of a genus 2 surface is linear, with
Ryan Budney.
Algebraic and Geometric Topology, 2001
A 64-dimensional faithful representation of
the mapping class group of a genus 2 surface.
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Braid groups are linear
Journal of the American Mathematical Society, 2001
The Lawrence-Krammer representation
is a faithful representation of the braid group
into a general linear group over the reals.
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Homological representations of braid groups
My PhD thesis, 2000.
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The Burau representation is not faithful for n=5
Geometry and Topology, 1999
The Burau representation of the braid group B_n
was shown to be unfaithful for n>5 by Long and Paton.
We use the same techniques to extend this to n=5.
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Supplements of bounded permutation groups
Journal of Symbolic Logic, 1998
Macpherson-Neumann classified a certain kind of infinite symmetric group,
but made a mistake in their proof.
I construct counterexamples in certain models of set theory,
and uncover a surprising connection to cardinal arithmetic.
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For completeness, here is my
honours thesis,
in tex, but untexable without some missing macros.
It is ancient history.