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

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 $SU(2)$-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 ${\mathcal{Q}}_0$.

Reorganized the paper, corrected typos and added more expositions on tight splitting. To appear in Geometry & Topology

References in corpus (6)

• Right-veering diffeomorphisms of compact surfaces with boundary II
• Khovanov homology detects the figure-eight knot
• Khovanov homology and the cinquefoil
• Small Dehn surgery and SU(2)
• Links all of whose cyclic branched covers are L-spaces
• Knot Floer homology, link Floer homology and link detection

Cited by in corpus (2)

• Closures of $3$-braids and detection
• Small Heegaard genus and SU(2)