Research

Papers: 

1. Non-Trivial Steenrod Squares on Prime Knots  

In this paper (preliminary version) we use ribbon concordances and their induced maps on Khovanov homology to show that there are prime knots with arbitrarily high non-trivial Steenrod Squares (defined by Lipshitz and Sarkar here: https://arxiv.org/abs/1204.5776). Applications include obstructions to Ribbon Concordance between two knots whose Khovanov stable homotopy type is known.

Here is a link to the published version: https://msp.org/agt/2022/22-3/p09.xhtml (Algebraic & Geometric Topology 22 (2022) 1273–1285)

I gave a talk about this work at SLMath (formerly MSRI) and it is available online here.

2. Obstructing reducible surgeries: Slice genus and thickness bounds. Joint with Robert DeYeso III. Preprint available here (submitted to MMJ)

Abstract: In this paper, we study reducible surgeries on knots in S^3. We develop thickness bounds for L-space knots that admit reducible surgeries, and lower bounds on the slice genus for general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots, which was mostly worked out in \cite{DeY21b}. We also provide a new upper bound on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots. Our techniques involve the d-invariants and mapping cone formula from Heegaard Floer homology. 

3. 3- and 4- dimensional invariants of satellite knots with (1,1) patterns. Preprint available here https://arxiv.org/abs/2305.18549 

In this paper I compute various topological and Floer theoretic invariants of satellite knots with (1,1)-patterns using the immersed curve reformulation of the bordered pairing theorem, in particular we compute \tau for an infinite family of satellite knots with arbitrary companions. Some applications to monodromy detection and number of minimal genus Seifert surfaces are discussed.

I gave a talk about this work in the Nearly Carbon Neutral Topology Seminar, available online here:

Part 1: https://youtu.be/Gp2IRWPTsF4?si=eRRTzaXXQGnxCRe1

Part 2: https://youtu.be/Xin_A2tczaY?si=GRn9uTlSuQlZw5-t

4. Genus, fiberedness, $\tau$ and $\epsilon$ of generalized Mazur patterns. Arxiv version https://arxiv.org/abs/2405.08763

We study a family of (1,1)-pattern knots that generalize the Mazur pattern, and compute the concordance invariants τ and ϵ of n-twisted satellites formed from these patterns. We show that none of the n-twisted patterns from this family act surjectively on the smooth or rational concordance group. We also determine when the n-twisted generalized Mazur patterns are fibered in the solid torus, compute their genus in the solid torus and show that n-twisted satellites with generalized Mazur patterns and non-trivial companions are not Floer thin. 

In Preparation

Reducible Surgeries on Almost L-space knots: Joint with Robert DeYeso III

Short Version: We study restrictions on reducing slopes on Almost L-space knots

Research Interests/Future Projects:

I am interested in applications of Khovanov homology to topological questions: concordance, non-orientable slice genus, and exotic surfaces.

I am also interested in applications of knot Floer homology to questions about Dehn surgery and the non-orientable slice genus. 

I study families of satellite knots with (1,1) patterns. One overall theme of this work is using the detection properties of knot Floer homology to detects topological aspects of pattern knots in the solid torus. An ongoing project is to determine what (if anything) the bordered module CFAhat(S^1\times D^2,P) has to say about genus and fiberedness of pattern knots.