Update
I am now an Assistant Professor at CSU Long Beach. For my new website click here.
Lecturer in Mathematics at UPenn
I study low dimensional geometry and topology with an emphasis on knot theory.
The world in which we live is three dimensional. This basic fact implies that we live in some 3-manifold. In this way, the study of 3-manifolds illuminates the nature of the universe. Knotting phenomena in manifolds is so rich that a complete understanding of it would lead to a complete understanding of all 3-manifolds. The broad goal of my research is to study 3-manifolds via knots.
My recent research investigates the structure of 3-manifolds and the knots they contain by applying modern techniques such as thin position and distance in the curve complex to classical constructions such as Heegaard splittings, Dehn surgery and diagrammatic knot invariants.
Genus Bounds Bridge Number for High Distance Knots(with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) Submitted
Bridge distance, Heegaard genus, and Exceptional Surgeries(with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) Submitted
High Distance Bridge Surfaces(with Maggy Tomova and Michael Yoshizawa) Submitted
Bridge Number and Tangle Products to appear in Algebraic & Geometric Topology
A Decomposition Theorem for Higher Rank Coxeter Groups(with Ryan Ottman) to appear in Communications in Algebra
Width is not Additive(with Maggy Tomova) to appear in Geometry & Topology
Companions of the Unknot and Width Additivity(with Maggy Tomova) to appear in J. Knot Theory Ramifications
Bridge Number and Conway Products Algebr. Geom. Topol. 10: 789-823 (electronic), 2010.
Alternating Augmentations of Links J. Knot Theory Ramifications, 18 (2009), 67-73.