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

TL;DR

This paper proves that certain genus-two fibered knots have fixed points in their pseudo-Anosov representatives, leading to the detection of the cinquefoil knot via knot Floer homology and establishing its uniqueness as a genus-two L-space knot.

Contribution

It demonstrates fixed points for pseudo-Anosov homeomorphisms of specific knots, proving knot Floer homology detects the cinquefoil and that it is the only genus-two L-space knot in $S^3$.

Findings

Knot Floer homology detects the cinquefoil knot T(2,5).

The cinquefoil is the only genus-two L-space knot in S^3.

All pseudo-Anosov homeomorphisms in certain strata have fixed points.

Abstract

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 $\mathcal{Q}_0$.