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.

Papers

 

Three-dimensional Rep-tiles (with Z. Marley, I. Richards) Submitted

Self-replicating 3-manifolds (with R. Lee) Submitted

The bridge number of arborescent links with many twigs (with S. Baader, A. Kjuchukova and F. Misev) Accepted to Algebraic and Geometric Topology

Kirby-Thompson distance for trisections of knotted surfaces (with M. Campisi, S. Taylor, M. Tomova) Accepted to Journal of the London Mathematical Society

A note on three-fold branched covers of S4 (with Patricia Cahn, Alexandra Kjuchukova and Jeffrey Meier) Accepted to Annales l'Institut Fourier

Coxeter groups and meridional rank of links (with Sebastian Baader and Alexandra Kjuchukova) Mathematische Annalen 379: 1533–1551, 2021

Knots with Exactly 10 Sticks (with Thomas Eddy, Nathaniel Morrison and Clayton Shonkwiler) Journal of Knot Theory and its Ramifications 29: 2050011, 2020

Distortion and the bridge distance of knots (with Marion Campisi, Scott Taylor and Maggy Tomova) Journal of Topology 13: 669-682, 2020

The incompatibility of crossing number and bridge number for knot diagrams (with Alexandra Kjuchukova and Makoto Ozawa) Discrete Mathematics 342: 1966-1978, 2019

Twist Number and the Alternating Volume of Knots (with Heidi Allen and Leslie Rodriguez) Journal of Knot Theory and its Ramifications 28: 1950016, 2019

Idempotents in Tangle Categories Split (with Joshua Sack) Journal of Knot Theory and its Ramifications 28: 1950025, 2019

Height, trunk and representativity of knots (with Makoto Ozawa) Journal of the Mathematical Society of Japan 71: 1105-1121, 2019

Wirtinger Systems of Generators of Knot Groups (with Alexandra Kjuchukova, Roman Velazquez, Paul Villanueva) Communications in Analysis and Geometry 28: 243-262, 2020

Dynamics of Embedded Curves by Doubly-Nonlocal Reaction-Diffusion Systems (with James von Brecht) Journal of Physics A: Mathematical on Theoretical 50: 1–57, 2017

Neighbors of knots in the Gordian graph (with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) The American Mathematical Monthly 124: 4--23, 2017

Exceptional and Cosmetic Surgeries on Knots(with Marion Campisi, Jesse Johnson, Scott Taylor and Maggy Tomova) Mathematische Annalen 367: 581--622, 2017

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.