(1) Constructing pseudo-Anosovs from expanding interval maps. To appear in Groups, Geometry, and Dynamics. [https://arxiv.org/abs/2101.01721]

I provide a characterization for a certain kind of interval map to be realizable as the train track map for a pseudo-Anosov. In particular, I show that, for each modality, such interval maps are parameterized by the rationals in the open unit interval. I also explicate a method due to W. Thurston by which one can realize the pseudo-Anosov from the action of a piecewise-affine lifting of the interval map on its limit set. I give necessary and sufficient conditions for this method to work, identifying it as providing explicit coordinates for a construction of A. de Carvalho.

(2) Fixed point-free pseudo-Anosovs and the cinquefoil, joint with Braeden Reinoso and Luya Wang. [https://arxiv.org/abs/2203.01402.]

We show that knot Floer homology detects the cinquefoil, i.e. the torus knot T(2,5). This paper builds on work of Baldwin-Hu-Sivek and many others, who reduced the problem to studying pseudo-Anosovs on the genus-2 surface with one boundary component that do not have any fixed points in the interior. The main new tool is the machinery of train tracks and a specialized kind of splitting, which allows us to study all pseudo-Anosovs by considering only a single train track. Our methods are of broad interest, and do not require a background in Floer theory.

(3) On the entropies of zig-zag maps of pseudo-Anosov type, in preparation.

I investigate open questions about the set of topological entropies of a class of interval maps which produce pseudo-Anosov homeomorphisms. The entropy of such an interval map coincides with that of the pseudo-Anosov it generates. The main tool in the analysis is cutting sequences of lines in the plane with rational slope, effectively imposing the structure of the Farey tree on the kneading sequences of the interval maps in question. In particular, I prove that entropy grows monotonically in the rational number associated to each interval map in (1). I also show that the closure of the set of Galois conjugates of exp(h(f)) contains the unit circle. The intersection of this set with the open unit disc corresponds to the set of roots of a certain class of power series with bounded coefficients. See below for images of this set.

(4) Spherical pseudo-Anosovs with one zero are interval-like, in preparation. Joint with Karl Winsor.

Let (S, w) be the data of the Riemann sphere equipped with a quadratic differential w. Suppose that w has a single zero, so that all other singularities of w are simple poles. Suppose further that f is a pseudo-Anosov homeomorphism of S preserving the horizontal and vertical foliations of w. We show that f is carried by a train track whose subgraph of expanding edges is homeomorphic to an interval. As an immediate consequence, we see that the dilatation of f is at least the square root of 2. This is a new proof of a result of Boissy and Lanneau, and provides a uniform lower bound on the systole of certain strata of moduli space, independent of the genus. Our techniques are a specialized kind of splitting, suited to standardly embedded train tracks, and was first introduced by Farber-Reinoso-Wang.

Here is a list of the upcoming conferences and seminars where I will be speaking. Come say hello!

1. The 55th Spring Topology and Dynamical Systems Conference. Friday, March 11.

2. AMS Spring Eastern Virtual Sectional Meeting: Special Session on Geometric Dynamics and Billiards. Sunday, March 20.

3. Boston College's Topology Reading Group. Thursday, March 24.

4. GSTGC 2022. April 1-3. Exact date TBD.

5. GT GAPS. Thursday, April 21.

6. Wash U Geometry and Topology Seminar. Friday, April 29.

I am currently reading several related papers. Key phrases include "Thurston norm," "mapping torus," and "fibered 3-manifold."

1. McMullen, Curtis T. Polynomial invariants for fibered 3-manifolds and Teïchmuller geodesics for foliations.

2. Thurston, William P. A norm for the homology of 3-manifolds.

3. Michael Landry, Yair Minsky, and Samuel J. Taylor. A polynomial invariant for veering triangulations.

4. Agol, Ian. Ideal triangulations of pseudo-Anosov mapping tori.