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Floer homology and right-veering monodromy

John A. Baldwin, Yi Ni, Steven Sivek

TL;DR

The paper proves that for a fibered knot K with monodromy h, the knot Floer complex detects whether h is right-veering via the invariant b(K), establishing b(K)>1 iff h is right-veering. The core method blends knot Floer theory with symplectic Floer homology, using a new criterion that HF^{symp}(h∪g_+) and HF^{symp}(h∪g_−) differ precisely when h is right-veering, and translating this into a 0-surgery framework to deduce the main result. This leads to a knot Floer–theoretic characterization of tight contact structures and yields several applications: constraints on monodromies of slice and persistently foliar knots, Dehn-surgery implications, and partial progress on the L-space conjecture. The work also demonstrates a concrete bridge between Heegaard Floer theory, symplectic Floer homology, and contact geometry, with potential independent interest in mapping-class dynamics. Overall, the paper provides a deep, technical link between fibered knot invariants and the dynamics of monodromy, yielding both theoretical and geometric consequences.

Abstract

We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering. In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda, Kazez, and Matic. Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms. We describe applications of this work to Dehn surgeries and taut foliations.

Floer homology and right-veering monodromy

TL;DR

The paper proves that for a fibered knot K with monodromy h, the knot Floer complex detects whether h is right-veering via the invariant b(K), establishing b(K)>1 iff h is right-veering. The core method blends knot Floer theory with symplectic Floer homology, using a new criterion that HF^{symp}(h∪g_+) and HF^{symp}(h∪g_−) differ precisely when h is right-veering, and translating this into a 0-surgery framework to deduce the main result. This leads to a knot Floer–theoretic characterization of tight contact structures and yields several applications: constraints on monodromies of slice and persistently foliar knots, Dehn-surgery implications, and partial progress on the L-space conjecture. The work also demonstrates a concrete bridge between Heegaard Floer theory, symplectic Floer homology, and contact geometry, with potential independent interest in mapping-class dynamics. Overall, the paper provides a deep, technical link between fibered knot invariants and the dynamics of monodromy, yielding both theoretical and geometric consequences.

Abstract

We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering. In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda, Kazez, and Matic. Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms. We describe applications of this work to Dehn surgeries and taut foliations.
Paper Structure (13 sections, 19 theorems, 180 equations, 4 figures)

This paper contains 13 sections, 19 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Proof outline
  4. Organization
  5. Acknowledgements
  6. Preliminaries
  7. Right-veering homeomorphisms
  8. Fractional Dehn twist coefficients
  9. Symplectic Floer homology
  10. Knot Floer homology and $b$
  11. The infinity knot Floer complex
  12. The Heegaard Floer homology of $0$-surgery
  13. Theorem \ref{['thm:main']} and its corollaries

Key Result

Theorem 1.1

A contact 3-manifold $(Y,\xi)$ is tight if and only if every fibered knot $K\subset Y$ supporting $(Y,\xi)$ has right-veering monodromy.Technically, Honda--Kazez--Matić's result says that $(Y,\xi)$ is tight iff every fibered link supporting $(Y,\xi)$ has right-veering monodromy, but this implies the

Figures (4)

  • Figure 1: $a$ is to the right of $b$ in a neighborhood of $p$.
  • Figure 2: A portion of $\Sigma$ near the multitwist region $R$ in the case $k=2$ and $\epsilon = +1$. That is, $R$ is made up of three positive twist regions, $A_1,A_2,A_3$, shaded in medium gray, together with the two fixed annuli between them, shaded in dark gray. The green arc is the image of the red under $\phi$. In this example, we see that $c_B(\varphi) \in (2,3)$.
  • Figure 3: The composition $h=D_x \circ D_{y}^{-1}$ of a right-handed Dehn twist about $x$ with a left-handed Dehn twist about $y$ is a non-right-veering homeomorphism of the once-punctured torus $S$. The arcs $a$ and $b$ form a basis for $S$, and are both moved to the right by $h$.
  • Figure 4: The standard representatives $\phi_\pm$ on the top and bottom, respectively. The green arcs are the images of the red arcs under the corresponding maps. The fixed annuli are shown in dark gray, and the twist regions in medium gray. Note that $\phi_+$ has one more fixed annulus $A$ than $\phi_-$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 36 more