arXiv subjects

Ethan Farber

Publications and source records attributed to Ethan Farber.

3 recordsLinked to original sources

arXiv2023

A Farey tree structure on a family of pseudo-Anosovs

We introduce a new perspective on a procedure for generating pseudo-Anosov homemorphisms from postcritically finite interval maps. The central idea is the realization of a tree structure on one such family of pseudo-Anosovs: individual systems, represented by the rational number measuring their rotation at infinity, are the vertices of the tree, while the edges encode dynamical relations between them. We also deepen the dictionary between one-, two- and three-dimensional invariants associated with these systems.

math.GT↗

arXiv2022

Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil

Given any genus-two, hyperbolic, fibered knot in $S^3$ with nonzero fractional Dehn twist coefficient, we show that its pseudo-Anosov representative has a fixed point. Combined with recent work of Baldwin--Hu--Sivek, this proves that knot Floer homology detects the cinquefoil knot $T(2,5)$, and that the cinquefoil is the only genus-two L-space knot in $S^3$. Our results have applications to Floer homology of cyclic branched covers over knots in $S^3$, to $\mathit{SU}(2)$-abelian Dehn surgeries, and to Khovanov and annular Khovanov homology. Along the way to proving our fixed point result, we describe a small list of train tracks carrying all pseudo-Anosov homeomorphisms in most strata on the punctured disk. As a consequence, we find a canonical track $\tau$ carrying all pseudo-Anosov homeomorphisms in a particular stratum $\mathcal{Q}_0$ on the genus-two surface, and describe every fixed-point-free pseudo-Anosov homeomorphism in $\mathcal{Q}_0$.

math.GT↗

arXiv2022

Constructing pseudo-Anosovs from expanding interval maps

We investigate a phenomenon observed by W. Thurston wherein one constructs a pseudo-Anosov homeomorphism on the limit set of a certain lift of a piecewise-linear expanding interval map. We reconcile this construction with a special subclass of generalized pseudo-Anosovs, first defined by de Carvalho. From there we classify the circumstances under which this construction produces a pseudo-Anosov. As an application, we produce for each $g \geq 1$ a pseudo-Anosov $ϕ_g$ on the surface of genus $g$ that preserves an algebraically primitive translation structure and whose dilatation $λ_g$ is a Salem number.

math.DS↗