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Ribbon concordance of fibered knots and compressions of surface homeomorphisms

Ian Agol, Qiuyu Ren

Abstract

We prove that simplicial volume and dilatation are monotone under ribbon concordance between fibered knots in $S^3$, and that every fibered knot has only finitely many predecessors in the ribbon-concordance partial order, providing evidence for questions raised by Gordon. We also give an algorithm to enumerate, up to symmetries, all minimal compressions of a surface homeomorphism, extending a theorem of Casson--Long. This yields an algorithm to find all knots that are strongly homotopy-ribbon concordant to a given fibered knot in some homotopy $I\times S^3$. Our study of minimal compressions also provides an alternative perspective on results of Miyazaki concerning nonsimple fibered ribbon knots.

Ribbon concordance of fibered knots and compressions of surface homeomorphisms

Abstract

We prove that simplicial volume and dilatation are monotone under ribbon concordance between fibered knots in , and that every fibered knot has only finitely many predecessors in the ribbon-concordance partial order, providing evidence for questions raised by Gordon. We also give an algorithm to enumerate, up to symmetries, all minimal compressions of a surface homeomorphism, extending a theorem of Casson--Long. This yields an algorithm to find all knots that are strongly homotopy-ribbon concordant to a given fibered knot in some homotopy . Our study of minimal compressions also provides an alternative perspective on results of Miyazaki concerning nonsimple fibered ribbon knots.
Paper Structure (16 sections, 21 theorems, 8 equations, 2 figures)

This paper contains 16 sections, 21 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ribbon concordances of fibered knots
  3. Compressions of surface homeomorphisms
  4. Organization of the paper
  5. Preliminaries
  6. Examples
  7. Monotonicity of simplicial volume
  8. Monotonicity of dilatation constant
  9. Algorithmic compressions of surface homeomorphisms
  10. Preliminaries and examples
  11. Finiteness of minimal compressions
  12. Canonical forms of minimal compressions
  13. Algorithm for minimal compressions
  14. Ribbon concordances of nonsimple fibered knots
  15. Example: The (2,1)-cable of the figure 8 knot
  16. ...and 1 more sections

Key Result

Theorem 1.4

If $K$ is fibered and $J\le K$, then $||S^3\backslash J||\le||S^3\backslash K||$.

Figures (2)

  • Figure 1: Plumb the square knot onto the figure $8$ knot, yielding a ribbon concordance from the figure $8$ knot (to the knot $9_{24}$ or its mirror).
  • Figure 2: The disks $D,D'$ with their double arcs and boundary data in the case $|\gamma\cap\delta|=4$. The components $\Sigma_1,\Sigma_2$ are pseudo-Anosov, and the components $\Sigma_1',\Sigma_1"$ are periodic.

Theorems & Definitions (39)

  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: casson1983loop
  • Theorem 1.8: casson1985algorithmic
  • Theorem 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Remark 1.12
  • Theorem 1.13
  • ...and 29 more