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

TL;DR

The paper establishes a geometric bridge between fixed-point properties of genus-two pseudo-Anosov monodromies and knot Floer homology, proving that nonzero fractional Dehn twist coefficients force fixed points and thereby enabling knot Floer detection of the cinquefoil T(2,5). Central to the argument is a two-case analysis of singularity types, combined with a robust train-track framework and a tight-splitting theory that yields canonical representations for pseudo-Anosovs in key strata. Consequences include that T(2,5) is the only genus-two L-space knot in S^3, and that T(2,5) is the sole genus-two knot with certain instanton and Khovanov detection properties; the results extend to Floer homology of branched covers and to annular Khovanov homology via the Birman–Hilden correspondence. The methods provide a versatile toolkit for future studies of dilatation spectra and fixed-point phenomena across surface dynamics and low-dimensional topology.

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 $\mathcal{Q}_0$.

Figures (22)

• Figure 1: Top (left to right): a 3-pronged saddle, a 4-pronged saddle, and a 1-pronged saddle at a marked point. Bottom (left to right): the neighborhood of a boundary singularity, and a 3-pronged boundary which permutes the first prong to the second.
• Figure 2: Left: the fibered surface for some geometric $\psi$ on $S_{0,5}^1$. The shaded regions are the junctions, and the striped bands connecting them are the strips. Right: following Bestvina-Handel, one inserts additional, "infinitesimal" edges into the junctions. These will also be inserted into the graph $G$ that one obtains by collapsing all of the decomposition elements. Their inclusion will produce the smooth analog $\tau$ of $G$, which is a train track. See Figure \ref{['fig:trackex']} below.
• Figure 3: A train track $\tau$ on the five-punctured disk $D_5$. The components of the complement of $\tau$ consist of: five once-punctured monogons, i.e. disks with a single boundary cusp; a trigon, i.e. a disk with three boundary cusps; and an exterior once-punctured bigon. Pseudo-Anosovs carried by this track lie in the stratum $(2;1^5;3)$.
• Figure 4: Three edge-paths on the train track $\tau$ from Figure \ref{['fig:trackex']}. Left: an edge path of length 7 which is not a train path, since it makes several sharp turns. Middle: a train path of length 7 which can be "pushed off" of $\tau$ into a small neighborhood so that it does not intersect itself. Right: a train path of length 6 which cannot be "pushed off" of $\tau$ so that it becomes injective.
• Figure 5: Top row: the train track $\tau$ from Figure \ref{['fig:trackex']} and the action of a pseudo-Anosov $\psi$ that it carries. The real edges of $\tau$ are labeled $e_1, \ldots, e_5$. The shaded regions on the right denote the neighborhoods that deformation retract onto these edges. Bottom three rows: The action of $f=(\text{collapse} \circ \psi)$ on each edge of $\tau$, depicted separately.

Theorems & Definitions (101)

• Theorem A
• Definition 1.1
• Theorem B
• Theorem C: cf. Theorem \ref{['thm:221']}
• Theorem D
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Remark 1.6
• Corollary 1.7