Pseudo-Anosov maps and genus-two L-space knots

Abstract

We classify genus-two L-space knots in S3 and the Poincare homology sphere.This leads to the first and to-date only detection results in knot Floer homology for knots of genus greater than one. Our proofs interweave Floer-homological properties of L-space knots, the geometry of pseudo-Anosov maps, and the theory of train tracks and folding automata for braids. The crux of our argument is a complete classification of fixed-point-free pseudo-Anosov maps in all but one stratum on the genus-two surface with one boundary component. To facilitate our classification, we exhibit a small family of train tracks carrying all pseudo-Anosov maps in most strata on the marked disk. As a consequence of our proof technique, we almost completely classify genus-two, hyperbolic, fibered knots with knot Floer homology of rank 1 in their next-to-top grading in any 3-manifold. Several corollaries follow, regarding the Floer homology of cyclic branched covers, SU(2)-abelian Dehn surgeries, Khovanov and annular Khovanov homology, and instanton Floer homology.