Tori and Heegaard splittings

Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface increases from zero, more complicated phenomena occur. Kobayashi showed that if a 3-manifold $M^3$ contains an essential torus $T$, then it contains one which can be isotoped to intersect a strongly irreducible Heegaard splitting surface $F$ in a collection of simple closed curves which are essential in $T$ and in $F$. In general there is no global bound on the number of curves in this collection. We give conditions under which a global bound can be obtained

Dehn surgery on complicated fibered knots in the 3-sphere

Let K be a fibered knot in the 3-sphere. We show that if the monodromy of K is sufficiently complicated, then Dehn surgery on K cannot yield a lens space. Work of Yi Ni shows that if K has a lens space surgery then it is fibered. Combining this with our result we see that if K has a lens space surgery then it is fibered and the monodromy is relatively simple

On tunnel number one knots that are not (1,n)

We show that the bridge number of a $t$ bridge knot in $S^3$ with respect to an unknotted genus $t$ surface is bounded below by a function of the distance of the Heegaard splitting induced by the $t$ bridges. It follows that for any natural number $n$, there is a tunnel number one knot in $S^3$ that is not $(1,n)$.Comment: 7 page

Knots and k-width

We investigate several integer invariants of curves in 3-space. We demonstrate relationships of these invariants to crossing number and to total curvature

Community Art Installation As An Approach To Art Therapy: A Community Engagement

In this thesis capstone paper, I advocate for a community art installation as an ethical and engaging way to approach art therapy. I argue that this approach may blur the lines between art therapy, art education and arts activism, but it is a powerful and anti-oppressive way to facilitate healing in marginalized communities. I found this approach furthered my understanding of what it means to be a competent and ethical art therapist. I also found it empowered art therapy group members and made them active facilitators of change in their lives and their communities. Observing the success of this project suggests a paradigm shift in approaches to art therapy

Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the boundary-connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.Comment: 10 page