Research

I study knots in 3-manifolds, surfaces in 4-manifolds, and the interactions between them in order to better understand the "weirdness" of 4-dimensional topology.  

Most of my work can be thought of answering one or both of the following broad questions:

1) How do algebraic invariants constrain and determine topological structure in 4-dimensions?

2) To what extent do associated manifolds and their invariants characterize the 4-dimensional properties of knots?

My favorite source of examples is the satellite construction, illustrated below, and I frequently rely on concordance invariants coming from metabelian representations of the knot group, like Casson-Gordon signatures and certain twisted Alexander polynomials. 

A pattern P in a solid torus

A companion knot K

The satellite knot P(K)

Preprints 

Publications

I contributed to the recent book The Disc Embedding Theorem, edited by Stefan Behrens, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray, coauthoring Chapter 3: "The Schoenflies theorem after Mazur, Morse, and Brown" with Stefan Behrens, Matthias Nagel, and Peter Teichner, and Chapter 18: "Tower height raising and embedding" with Mark Powell. 

 Stefan Friedl, Mark Powell, and I wrote a short note titled Linking forms of amphichiral knots  in 2017, before learning that our main corollary was proven by Goeritz in 1933. 


Code and computations: