Publications

• Skein theory for the ADE planar algebras, (preprint).
  Gives generators and relations for the ADE planar algebras.
• Generalized Long-Moody representations of braid groups, with Jianjun Tian
  Communications in Contemporary Mathematics (2008)
  Algorithms to build new representations from old.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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).
• 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.
• 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.
• 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.
• Homological representations of braid groups
  My PhD thesis, 2000.
• 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.
• 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.
• For completeness, here is my honours thesis, in tex, but untexable without some missing macros. It is ancient history.