From 2011 to 2013 I was a Postdoctoral Research
Fellow at the University of Western
Sydney where my supervisor was
Volker Gebhardt. I completed my
Ph.D. at the
University of Warwick in 2009 where
my supervisor was Daan Krammer and my
thesis was on plat closure of braids. I'm interested in the
algebraic and computational properties of braid groups and related
groups and using these properties to address problems in geometric
group theory, knot theory and low-dimensional topology.

Research interests

Braid groups, Garside groups, mapping class groups, geometric group
theory, low-dimensional topology

We prove that the exponential growth rate of the regular
language of penetration sequences is smaller than the
growth rate of the regular language of normal form
words, if the acceptor of the regular language of normal
form words is strongly connected. Moreover, we show that
the latter property is satisfied for all irreducible
Artin monoids of spherical type, extending a result by
Caruso.

Apart from establishing that the expected value of the
penetration distance $\mathrm{pd}(x,y)$ in irreducible
Artin monoids of spherical type is bounded independently
of the length of $x$, if $x$ is chosen uniformly among
all elements of given canonical length and $y$ is chosen
uniformly among all atoms, our results also give an
affirmative answer to a question posed by Dehornoy.
arXiv
MathSciNet

A monoid $K$ is the internal Zappa-Szép product of two
submonoids, if every element of $K$ admits a unique
factorisation as the product of one element of each of
the submonoids in a given order. This definition yields
actions of the submonoids on each other, which we show
to be structure preserving. We prove that $K$ is a
Garside monoid if and only if both of the submonoids are
Garside monoids. In this case, these factors are
parabolic submonoids of $K$ and the Garside structure of
$K$ can be described in terms of the Garside structures
of the factors. We give explicit isomorphisms between
the lattice structures of $K$ and the product of the
lattice structures on the factors that respect the
Garside normal forms. In particular, we obtain explicit
natural bijections between the normal form language of
$K$ and the product of the normal form languages of its
factors.
arXiv
MathSciNet

Analysing statistical properties of the normal forms of
random braids, we observe that, except for an initial
and a final region whose lengths are uniformly bounded
(that is, the bound is independent of the length of the
braid), the distributions of the factors of the normal
form of sufficiently long random braids depend neither
on the position in the normal form nor on the lengths of
the random braids. Moreover, when multiplying a braid on
the right, the expected number of factors in its normal
form that are modified, called the expected penetration
distance, is uniformly bounded.

We explain these observations by analysing the growth
rates of two regular languages associated to normal
forms of elements of Garside groups, respectively to the
modification of a normal form by right multiplication.

A universal bound on the expected penetration distance
in a Garside group yields in particular an algorithm for
computing normal forms that has linear expected running
time.
slidesarXiv
MathSciNet

Consider the unit ball, $B = D \times [0,1]$, containing
$n$ unknotted arcs $a_1, a_2, \ldots, a_n$ such that the
boundary of each $a_i$ lies in $D \times \{0\}$. The
Hilden (or Wicket) group is the mapping class group of
$B$ fixing the arcs $a_1 \cup a_2 \cup \cdots \cup a_n$
setwise and fixing $D \times \{1\}$ pointwise. This
group can be considered as a subgroup of the braid
group. The pure Hilden group is defined to be the
intersection of the Hilden group and the pure braid
group.

In a previous paper we computed a presentation for the Hilden
group using an action of the group on a cellular complex. This
paper uses the same action and complex to calculate a finite
presentation for the pure Hilden group. The framed braid group
acts on the pure Hilden group by conjugation and this action is
used to reduce the number of cases.
pdfarXiv
MathSciNet

Erratum: A presentation for Hilden's subgroup of the braid group

After publication of my previous paper Allen Hatcher
found a gap in the proof that the complex $X_n$ is
simply connected. This complex is defined in terms of
isotopy classes of discs, but the argument uses
representatives of the isotopy classes. There was an
implicit assumption that for an edge path in the complex
there exists sufficiently nice representatives of each
isotopy class. In this paper the properties of these
representatives will be made explicit. It is clear that
such representatives exist for a path, the problem is
that for a loop it is not obvious that the
representative at the beginning and end can be chosen to
coincide. This paper addresses this problem and contains
the complete proof that $X_n$ is simply connected,
incorporating all of the necessary changes. There were
also small errors in Figure 7 and Figure 8 and the
correct versions of these are included.
pdf
MathSciNet

A presentation for Hilden's subgroup of the braid group

Consider the unit ball, $B = D \times [0,1]$, containing
$n$ unknotted arcs $a_1, \ldots, a_n$ such that the
boundary of each $a_i$ lies in $D \times \{0\}$. We
give a finite presentation for the mapping class group
of $B$ fixing the arcs $\{ a_1, \ldots, a_n\}$ setwise
and fixing $D \times \{1\}$ pointwise. This
presentation is calculated using the action of this
group on a simply-connected complex.
pdfarXiv
MathSciNet

A tool to automatically produce pictures of elements of the braid
and Hilden groups.

Implementing a solution to the generalised word problem for the Hilden group

Abstract

Saul Schleimer gave a solution to the membership problem
of the handlebody mapping class group in the surface
mapping class group. We adapt this to solve the
membership problem of the Hilden group in the braid
group and implement this algorithm in
Magma.
pdfMagmasrc