[Paper Review] Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil

Ethan Farber, Braeden Reinoso+1 more|arXiv (Cornell University)|Mar 2, 2022

Geometric and Algebraic Topology5 citations

TL;DR

This paper proves that every genus-two, hyperbolic, fibered knot in $S^3$ with nonzero fractional Dehn twist coefficient has a pseudo-Anosov representative with a fixed point, thereby establishing that knot Floer homology detects the cinquefoil knot $T(2,5)$ and that it is the only genus-two L-space knot in $S^3$. The proof relies on train track dynamics and a novel algorithm for tight splitting of train tracks to analyze fixed-point-free pseudo-Anosov maps.

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 $τ$ 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 $\mathcal{Q}_0$.

Motivation & Objective

• To resolve the open question of whether knot Floer homology detects the torus knot $T(2,5)$, the cinquefoil.
• To prove that every genus-two, hyperbolic, fibered knot in $S^3$ with nonzero fractional Dehn twist coefficient has a pseudo-Anosov representative with a fixed point.
• To develop a systematic method for analyzing dilatations of pseudo-Anosov maps in genus two using train tracks.
• To apply these results to detect L-space knots and understand branched covers and $ ext{SU}(2)$-abelian surgeries.

Proposed method

• Uses train track theory to model pseudo-Anosov maps, focusing on a canonical track $\tau$ in the stratum $\mathcal{Q}_0$ on the genus-two surface.
• Introduces a tight splitting algorithm that reduces the number of joints in a train track by iteratively splitting at non-rigid loop switches of maximal valence.
• Applies Perron-Frobenius theory to transition matrices associated with train tracks to ensure finite termination of the splitting process.
• Leverages the hyperelliptic involution to quotient pseudo-Anosov maps on punctured surfaces to braids, enabling analysis via standard train tracks like the Peacock and Snail.
• Employs a joint-reduction algorithm that tracks transition matrices and eigenvectors to guarantee that all stems are eventually split, ensuring termination.
• Uses the fact that any pseudo-Anosov in the stratum $(4;\emptyset;3^2)$ is conjugate to one carried by a specific train track, reducing the problem to a finite combinatorial search.

Experimental results

Research questions

• RQ1 Does knot Floer homology detect the cinquefoil knot $T(2,5)$ among genus-two knots in $S^3$?
• RQ2 Can a genus-two, hyperbolic, fibered knot in $S^3$ admit a fixed-point-free pseudo-Anosov representative if its fractional Dehn twist coefficient is nonzero?
• RQ3 What is the complete set of train tracks that carry all pseudo-Anosov maps in a given stratum of the genus-two surface?
• RQ4 How can the dilatation spectrum of pseudo-Anosov maps in genus two be systematically studied using train track dynamics?
• RQ5 Can the structure of branched covers of knots in $S^3$ be constrained using dynamical properties of their monodromies?

Key findings

• Any genus-two, hyperbolic, fibered knot in $S^3$ with nonzero fractional Dehn twist coefficient must have a pseudo-Anosov monodromy with a fixed point in the interior of the fiber surface.
• The cinquefoil knot $T(2,5)$ is the only genus-two L-space knot in $S^3$, as established by combining the fixed-point result with prior work of Baldwin–Hu–Sivek.
• All pseudo-Anosov maps in the stratum $(4;\emptyset;3^2)$ on the genus-two surface with one boundary component are conjugate to maps carried by a specific train track, which is the lift of the Peacock track on the punctured disk.
• The algorithm for tight splitting of train tracks with joints terminates in finitely many steps, reducing the number of joints to zero and ensuring that all real edges are eventually split over.
• Every pseudo-Anosov map on the punctured disk with a $k$-pronged singularity ($k \geq 2$) away from the boundary is carried by a standardly embedded train track with no joints.
• The transition matrix associated with a train track and a pseudo-Anosov map has a unique, positive, simple Perron-Frobenius eigenvalue equal to the dilatation of the map.