Research

Generally I work in the field of low dimensional topology. I am interested in knot theory, braid groups, and other visual/combinatorial objects. I like to study maps between things, so I am naturally drawn to combinatorial representation theory, finite type invariants, planar algebras, and quantum computing.

The topic of my thesis was finding discrete representations of the braid groups into lattices. My approach was to specialize the parameter of the representation to certain Salem numbers, and harness the algebraic properties of the Salem numbers to guarantee discreteness of the representation.

Since graduating, I have focused my research towards virtual knot theory and more general studies of the braid groups.

Here is my CV.

Here is a link to my ArXiv papers.

Here are my papers:

The quotients of the Braid Groups by Their Congruence Subgroups  joint with Wade Bloomquist and Peter Patzt

 

Large 1-Systems of Curves in Non-Orientable Surfaces joint with students Sarah Ruth Nicholls and Julie Shneidman as part a summer REU project.

 

Yarn Ball Knots and Faster Computations, joint with Dror Bar-Natan, Itai Bar-Natan and Iva Halecheva

 

 

 

An Invariant of Virtual Trivalent Spatial Graphs, joint with Sherilyn Tamagawa

 

 

Finite Image Homomorphisms of the Braid Group and its Generalizations  Joint with Yvon Veberne

 

   OC relation— A quick proof of a known fact that the OC  relation can be written in two ways.

 

Ribbon 2-Knots, 1+1=2, and Duflo’s Theorem for Arbitrary Lie Algebras  joint with Dror Bar-Natan and Zsuzsanna Dancso

 

 

PhD Thesis: Discrete Representations of the Braid Groups

 

 

 

Complete Classification of Discrete Specializations of the Burau Representation of B_3

 

 

 

A Survey of Grid Diagrams and a Proof of Alexander’s Theorem

 

 

My Master’s thesis: The Alexander Polynomial

 

 

Turning Math Into Dance: Lessons from Dancing my PhD