# Stephen Tawn

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

## Publications

### Constructing unlabelled lattices

Volker Gebhardt
Abstract

We present an improved orderly algorithm for constructing all unlabelled lattices up to a given size, that is, an algorithm that constructs the minimal element of each isomorphism class relative to some total order.

Our algorithm employs a stabiliser chain approach for cutting branches of the search space that cannot contain a minimal lattice; to make this work, we grow lattices by adding a new layer at a time, as opposed to adding one new element at a time, and we use a total order that is compatible with this modified strategy.

The gain in speed is between one and two orders of magnitude. As an application, we compute the number of unlabelled lattices on 20 elements.

### On the penetration distance in Garside monoids

Volker Gebhardt
Abstract

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.

### Zappa-Szép products of Garside monoids

Volker Gebhardt
Abstract

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.

### Normal forms of random braids

Volker Gebhardt
Abstract

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. slides

### A presentation for the pure Hilden group

Abstract

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. pdf

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

Abstract

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

### A presentation for Hilden's subgroup of the braid group

Abstract

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. pdf

## Other

### Draw a braid

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.