My Research

 

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.

Submitted Papers

 

Neighbors of knots in the Gordian graph (with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) To appear in The American Mathematical Monthly

Exceptional and Cosmetic Surgeries on Knots(with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) To appear in Mathematische Annalen

Knots with Compressible Thin Levels (with Alexander Zupan) Algebraic and Geometric Topology 15: 1691--1717 (electronic), 2015

Essential Surfaces in Highly Twisted Link Complements (with David Futer and Maggy Tomova) Algebraic and Geometric Topology 15: 1501--1525 (electronic), 2015

Distance Two Links (with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) Geometriae Dedicata 7: 1--21, 2015.

A Prime Decomposition Theorem for the String Link Monoid (with R. Koytcheff and J. Burke) Journal of Knot Theory and its Ramifications, 24: 1--24, 2015.

High Distance Bridge Surfaces(with Maggy Tomova and Michael Yoshizawa) Algebraic and Geometric Topology 13: 2925--2946 (electronic), 2013.

Bridge Number and Tangle Products Algebraic and Geometric Topology 13: 1125--1141 (electronic), 2013.

A Decomposition Theorem for Higher Rank Coxeter Groups(with Ryan Ottman) Communications in Algebra, 20: 2508--2518, 2013.

Width is not Additive(with Maggy Tomova) Geometry & Topology, 17: 93--156, 2013.

Companions of the Unknot and Width Additivity(with Maggy Tomova) Journal of Knot Theory and its Ramifications, 20: 497--511, 2011.

Bridge Number and Conway Products Algebraic and Geometric Topology 10: 789--823 (electronic), 2010.

Alternating Augmentations of Links Journal of Knot Theory and its Ramifications, 18 (2009), 67--73.