Ryan Blair

COMPANIONS OF THE UNKNOT AND WIDTH ADDITIVITY

It has been conjectured that for knots K and K′ in S3, w(K # K′) = w(K) + w(K′) - 2. In [7], Scharlemann and Thompson proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots

Neighbors of Knots in the Gordian Graph

\{K^n_i\}

A prime decomposition theorem for the 2-string link monoid

for which they conjectured that

A Decomposition Theorem for Higher Rank Coxeter Groups

w(B^n\, \#\, K^n_i) = w(K^n_i)

ALTERNATING AUGMENTATIONS OF LINKS

where Bn is a bridge number n knot. We show that for n > 2 none of the knots in

High distance bridge surfaces

\{K^n_i\}

Bridge distance, Heegaard genus, and Exceptional Surgeries

produces such counterexamples.

Width is not additive

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.

Bridge number and Conway products

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.

Wirtinger systems of generators of knot groups

In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group where the groups in question do not necessarily act cocompactly or cofinitely. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups and to produce structure theorems for hyperbolic Coxeter groups.