Research:

I study 3dimensional
topology, especially knots, links,
and hyperbolic 3manifolds. My advisor was Alan Reid
at the University of Texas.
Papers
and Preprints:
 Closed surfaces and character
varieties. Algebraic
&
Geometric
Topology, Vol. 13
(2013) 20012037.
We show that
module structures on the coordinate
ring of the (P)SL(2,C) character variety for a
knot manifold can be
used to identify when boundary slopes and
closed essential surfaces are
detected by the techniques of Culler and
Shalen. The paper includes
numerous examples.

 Some Virtually Special
Hyperbolic 3Manifold Groups
(with Jason
Deblois
and Henry
Wilton). Commentarii
Mathematici
Helvetici Vol. 87 (2012), 727787.
We show that
hyperbolic 3manifolds that admit a
decomposition into rightangled ideal
polyhedra are virtually fibered
and LERF. The paper includes numerous
examples.

 Algebraic invariants,
mutation, and commensurability of link complements
(with Jason
Deblois). Pacific
Journal of Mathematics Accepted for publication.
We
construct an infinite family of hyperbolic
twocomponent links and
investigate their geometric and
commensurability properties.
Among other things, we show that mutants
of these manifolds
produce arbitrarily large finite subfamilies
of nonisometric manifolds
with the same volume and scissors congruence
class. Depending
on
the choice of mutation, these manifolds may be
commensurable or
incommensurable.

 Not all boundary
slopes are strongly detected by the character variety
(with Stephan
Tillmann). Communications
in Analysis and Geometry Vol. 15
(2007), no. 4.
We
answer the question of whether all boundary
slopes of a hyperbolic
3manifold are strongly detected by the
character variety by giving an
infinite family of hyperbolic links which have
boundary slopes that are
not strongly detected.

 All roots of unity
are detected by the Apolynomial. Algebraic
&
Geometric
Topology, Vol. 5
(2005) 207217.
We answer a
question of Cooper, Culler, Gillet,
Long, and Shalen as to
which roots of unity can arise when a boundary
slope is strongly
detected by the character variety. For
each positive integer
n, we give
examples of infinitely many hyperbolic
manifolds where every nth root
of unity arises in this process.

 Undetected boundary
slopes and roots of unity for the character variety of
a 3manifold.
