My publications as of 2020
"Supplements of bounded permutation groups",
Journal of Symbolic Logic, 1998.
"The Burau representation is not faithful for n=5",
Geometry and Topology, 1999.
"Braid groups are linear",
Journal of the American Mathematical Society, 2000.
I contributed to a problem list.
"The mapping class group of a genus two surface is linear",
with Ryan Budney,
Algebraic and Geometric Topology, 2001.
"Does the Jones polynomial detect the unknot?",
Journal of Knot Theory and its Ramifications, 2002.
"A homological definition of the Jones polynomial",
Geometry and Topology Monographs, 2002.
"Representations of braid groups",
Proceedings of the International Congress of Mathematics, 2002.
"The Lawrence-Krammer representation",
Topology and Geometry of Manifolds, 2003.
"Homological representations of the Iwahori-Hecke algebra",
Geometry and Topology Monographs, 2004.
"Braid groups and Iwahori-Hecke algebras",
Proceedings of Symposia in Pure Mathematics, 2006.
"A homological definition of the HOMFLY polynomial",
Algebraic and Geometric Topology, 2007.
"Generalized Long-Moody representations of braid groups",
with Jianjun Paul Tian,
Communications in Contemporary Mathematics, 2008.
"Skein theory for the ADE planar algebras",
Journal of Pure and Applied Algebra, 2010.
"A diagrammatic Alexander invariant of tangles",
Journal of Knot Theory and its Ramifications, 2012.
"The Alexander and Jones polynomials through representations of rook algebras",
with Eric Ramos and Ren Yi, JKTR, 2012.
"Constructing the extended Haagerup planar algebra",
with Scott Morrison, Emily Peters and Noah Snyder,
Acta Mathematica, 2012.
"Principal graph stability and the jellyfish algorithm",
with David Penneys,
Mathematische Annalen, 2014.
"A diagrammatic definition of Uqsl2",
Journal of Knot Theory and its Ramifications, 2014.
"Alexander representation of tangles",
with Alessia Cattabriga and Vincent Florens,
Acta Mathematica Vietnamica, May 2015,
"The pop-switch planar algebra and the Jones-Wenzl idempotents",
with Ellie Grano,
Journal of Knot Theory and its Ramifications, June 2015.
(A "featured article" for 2015!)
"Bowling ball representations of braid groups",
Journal of Knot Theory and its Ramifications, March 2018.
"An exact entangling gate Using Fibonacci anyons"
with Claire Levaillant,
Bulletin of the Australian Mathematical Society, November 2018.
An entry in The Concise Encyclopedia of Knot Theory
"On the Burau representation of B4 modulo p",
with Beridze and Traczyk.
"Review: the annular structure of subfactors, by Vaughan Jones".
I try hard to make my papers readable and correct,
but inevitably fall short.
I decided to keep a list of errors as I find them.
The Burau representation is not faithful for n=5:
I called the full twist braid Δ,
but the usual name is Δ-squared.
Braid groups are linear:
There is a sign error in Theorem 4.1,
which gives the matrix entries for the faithful representation.
Does the Jones polynomial detect the unknot?
I slightly regret the title,
since the paper is mostly about Burau-4.
Also I regret saying "In the case of B3,
I know of no other value of q other than roots of unity at which the Burau representation is unfaithful".
This was true,
but it was pointed out to me that such values are actually very easy to find.
A homological definition of the HOMFLY polynomial:
This is actually a homological definition of the SLn invariants,
but at least I confessed to this slight deception in the paper.
Generalized Long-Moody representations of braid groups:
Somebody pointed out an error but I have forgotten what it was –
a sign error, something acting on the wrong side, or something like that.
Skein theory for the ADE planar algebras:
The proof of Lemma 3.2 did not say why R generates P'.
(It's because R generates the minimal projectors.)
The proof of Lemma 4.5 has a greater-than symbol that should be less-than.
A diagrammatic definition of Uqsl2:
Section 6 constructs a quotient of H by a certain intersection of kernels.
However it is not clear that this intersection of kernels is a Hopf ideal.
Instead, you can quotient by the specific generators I want to get rid of
and notice that the result still has a well-defined Hopf algebra structure.
My student Vijay thinks he can fix my original approach.