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

Farber, Ethan¹ ; Reinoso, Braeden¹ ; Wang, Luya²

¹ Boston College, Chestnut Hill, MA, United States
² Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States, Institute for Advanced Study, Princeton, NJ, United States

Geometry & topology, Tome 28 (2024) no. 9, pp. 4337-4381

Résumé

Given any genus-two, hyperbolic, fibered knot in S3 with nonzero fractional Dehn twist coefficient, we show that its pseudo-Anosov representative has a fixed point. Combined with recent work of Baldwin, Hu and 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 S3. Our results have applications to Floer homology of cyclic branched covers over knots in S3, 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 τ carrying all pseudo-Anosov homeomorphisms in a particular stratum 𝒬0 on the genus-two surface, and describe every fixed-point-free pseudo-Anosov homeomorphism in 𝒬0.

Détail

DOI : 10.2140/gt.2024.28.4337

Keywords: pseudo-Anosov homeomorphism, knot Floer homology, train track, fibered knot, fixed point

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