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

Not all knots are smoothly round handle slice, joint with Tye Lidman and Arunima Ray, July 2025, 4 pages.
Slice disks modulo local knotting, joint with Jeffrey Meier, March 2025, 22 pages.
Immersed surfaces with knot group ℤ, joint with Anthony Conway, October 2024, 76 pages.
A partial resolution of Hedden's conjecture on satellite homomorphisms, joint with Randall Johanningsmeier (Swat '24) and Hillary Kim (Swat '25), August 2023, 11 pages, to appear in Mathematical Proceedings of the Cambridge Philosophical Society.

Publications

Linking number obstructions to satellite homomorphisms, joint with Tye Lidman and Juanita Pinzón-Caicedo, Quantum Topology (2024), published online first, 28 pages. arXiv version.
A note on surfaces in ℂℙ2 and ℂℙ2 # ℂℙ2, joint with Marco Marengon, Arunima Ray, and András I. Stipsicz, Proceedings of the American Mathematical Society, Series B 11 (2024), pp. 187-199. arXiv version.
Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group, joint with Mark Powell, Journal of the London Mathematical Society 107 no. 6 (2023), pp. 2025-2053. arXiv version.
Slicing knots in definite 4-manifolds, joint with Alexandra Kjuchukova, Arunima Ray, and Sümeyra Sakallı, Transactions of the American Mathematical Society 377 (2024), pp. 5905-5946. arXiv version.
A note on the concordance Z-genus, joint with JungHwan Park, Michigan Mathematical Journal 74 (1) (2024) pp. 73-83, DOI: 10.1307/mmj/20216070. arXiv version.
Amphichiral knots with large 4-genus, Bulletin of the London Mathematical Society 54 (2022), pp. 624-634. arXiv version.
Embedding spheres in knot traces, joint with Peter Feller, Matthias Nagel, Patrick Orson, Mark Powell, and Arunima Ray, Compositio Mathematica 157 no. 10 (2021), pp. 2242-2279. arXiv version.
Branched covers bounding rational homology balls, joint with Paolo Aceto, Jeffrey Meier, Maggie Miller, JungHwan Park, and András I. Stipsicz, Algebraic & Geometric Topology 21 no. 7 (2021), pp 3569–3599. arXiv version.
Homomorphism obstructions for satellite maps, Transactions of the American Mathematical Society, Series B 10 (2023), pp. 220-240. arXiv version.
Stabilization distance between surfaces, joint with Mark Powell, L’Enseignement Mathématique 65 no. 3/4 (2019), pp. 397-440. arXiv version.
A note on the topological slice genus of satellite knots, joint with Peter Feller and Juanita Pinzón-Caicedo, Algebraic & Geometric Topology 22 no. 2 (2022), pp. 709--738. arXiv version.
Two-solvable and two-bipolar knots with large four-genera, joint with Jae Choon Cha and Mark Powell, Mathematical Research Letters 28 no. 2 (2021), pp. 331-382. arXiv version.
Winding number m and -m patterns acting on concordance, Proceedings of the American Mathematical Society 147 (2019), pp. 2723-2731. arXiv version.
Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance, joint with Mark Powell, Algebraic & Geometric Topology 18 no. 6 (2018), 3425-3476. arXiv version.
Knot traces and concordance, joint with Lisa Piccirillo, Journal of Topology 11 no. 1 (2018), 201-220. arXiv version.
The topological sliceness of 3-strand pretzel knots, Algebraic & Geometric Topology 17 (2017) 3057–3079. arXiv version.
A note on the topological sliceness of some 2-bridge knots, Mathematical Proceedings of the Cambridge Philosophical Society 164 no. 1 (2017), 185-191. arXiv version.
Distinguishing mutant pretzel knots in concordance, Journal of Knot Theory and its Ramifications 26 no. 7 (2017), 24 pp. arXiv version.
Ravels arising from Montesinos tangles, joint with Erica Flapan, Tokyo Journal of Mathematics 40 no. 2 (2017), 393-420. arXiv version.
The Wecken property for random maps on surfaces with boundary, joint with Jacqueline Brimley, Matthew Griisser, and P. Christopher Staecker, Topology and its Applications 159 no. 18 (2012), 3662-676. arXiv version.

I also contributed to 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:

Sample Maple code for computing twisted Blanchfield pairings of the knot 8_8, as in the paper Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance is available here. (Both paper and code are joint work with Mark Powell.) If you are primarily interested in computing twisted Alexander polynomials rather than the full pairing, this program can be simplified and will run significantly faster- feel free to email if you have questions about how to do this!
The computations for the paper Branched covers bounding rational homology balls, joint with Paolo Aceto, Jeffrey Meier, Maggie Miller, JungHwan Park, and András Stipsicz are here: Alexander module and Blanchfield pairing computations (Jupyter notebook or pdf printout) and twisted Alexander polynomial computations (Maple worksheet or pdf printout).

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