Research:
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I study 3-dimensional topology, especially knots, links,
and hyperbolic 3-manifolds. My advisor was Alan Reid
at the University of Texas.
Papers and Preprints:
- Closed surfaces and character
varieties.
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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.
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- Some Virtually Special
Hyperbolic 3-Manifold Groups (with Jason Deblois and Henry Wilton).
Accepted for publication in Commentarii
Mathematici Helvetici.
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We show that hyperbolic 3-manifolds that admit a
decomposition into right-angled ideal polyhedra are virtually fibered
and LERF. The paper includes numerous examples.
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- Algebraic invariants,
mutation, and commensurability of link complements (with Jason Deblois).
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We
construct an infinite family of hyperbolic two-component 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.
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- Not all boundary
slopes are strongly detected by the character variety (with Stephan
Tillmann). Communications
in Analysis and Geometry Vol. 15
(2007), no. 4.
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We
answer the question of whether all boundary slopes of a hyperbolic
3-manifold are strongly detected by the character variety by giving an
infinite family of hyperbolic links which have boundary slopes that are
not strongly detected.
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- All roots of unity
are detected by the A-polynomial. Algebraic
&
Geometric Topology, Vol. 5 (2005) 207-217.
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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.
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- Undetected boundary
slopes and roots of unity for the character variety of a 3-manifold.
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