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Slice disks modulo local knotting

Jeffrey Meier, Allison N. Miller

TL;DR

The paper tackles the problem of classifying slice disks for a fixed knot up to LK-isotopy by constructing infinite families of disks that remain nonisotopic after local knotting. It develops a twist-spun disk framework to produce pairwise nonisotopic disks for $K=J\#\overline{J}$ and, for coprime $p,q>1$, a rich family of fibered, homotopy-ribbon disks bounded by the generalized square knot $Q_{p,q}$ with pairwise distinct LK-relations, all while exteriors are diffeomorphic. The work further extends these phenomena through invertible concordances and satellite operations, yielding many more infinite families of inequivalent disks, and it uses both geometric/group-theoretic and Alexander-module techniques to detect nonisotopy via kernel data and Blanchfield-type pairings. Collectively, these results advance understanding of the diversity of slice disks for a fixed knot and provide constructive methods to generate and distinguish large classes of inequivalent disks in both smooth and topological categories.

Abstract

The second author and Powell asked whether there exist knots bounding infinitely many slice disks that remain pairwise nonisotopic, even after local knotting. We answer this question in the affirmative, giving many classes of examples distinguished by the kernels of the inclusion-induced maps on the fundamental group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy modulo local knotting, extending work of the first author and Zupan. We conclude with a discussion of how invertible concordances and satellite operations can produce new examples of knots bounding many inequivalent slice disks. In particular, we give examples of fibered, hyperbolic knots bounding infinitely many fibered, ribbon disks that are pairwise nonisotopic modulo local knotting. The closed monodromies of these knots are pseudo-Anosov mapping classes that have infinitely many distinct handlebody extensions, a curiosity that may be of independent interest.

Slice disks modulo local knotting

TL;DR

The paper tackles the problem of classifying slice disks for a fixed knot up to LK-isotopy by constructing infinite families of disks that remain nonisotopic after local knotting. It develops a twist-spun disk framework to produce pairwise nonisotopic disks for and, for coprime , a rich family of fibered, homotopy-ribbon disks bounded by the generalized square knot with pairwise distinct LK-relations, all while exteriors are diffeomorphic. The work further extends these phenomena through invertible concordances and satellite operations, yielding many more infinite families of inequivalent disks, and it uses both geometric/group-theoretic and Alexander-module techniques to detect nonisotopy via kernel data and Blanchfield-type pairings. Collectively, these results advance understanding of the diversity of slice disks for a fixed knot and provide constructive methods to generate and distinguish large classes of inequivalent disks in both smooth and topological categories.

Abstract

The second author and Powell asked whether there exist knots bounding infinitely many slice disks that remain pairwise nonisotopic, even after local knotting. We answer this question in the affirmative, giving many classes of examples distinguished by the kernels of the inclusion-induced maps on the fundamental group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy modulo local knotting, extending work of the first author and Zupan. We conclude with a discussion of how invertible concordances and satellite operations can produce new examples of knots bounding many inequivalent slice disks. In particular, we give examples of fibered, hyperbolic knots bounding infinitely many fibered, ribbon disks that are pairwise nonisotopic modulo local knotting. The closed monodromies of these knots are pseudo-Anosov mapping classes that have infinitely many distinct handlebody extensions, a curiosity that may be of independent interest.

Paper Structure

This paper contains 16 sections, 19 theorems, 61 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Directions for future work
  3. Definitions
  4. Notation
  5. Definitions
  6. Twist-spun disks
  7. Disks for generalized square knots
  8. Seifert fibered structures via circle actions
  9. Presentations for the group of the torus knot
  10. Presentations for the group of the product ribbon disk
  11. The homotopy class of a slope on the pillowcase
  12. Identifying the kernels
  13. Other examples
  14. Invertible concordances
  15. Satellite operations
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $p,q>0$ be coprime. The generalized square knot $Q_{p,q} = T_{p,q}\#\overline T_{p,q}$ bounds infinitely many fibered, homotopy-ribbon disks that are pairwise nonisotopic modulo local knotting.

Figures (6)

  • Figure 1: The torus knot $J = T_{7,5}$ as an orbit of a circle-action on $S^3$. Here, the curve $\eta$ is a $(4,3)$--curve.
  • Figure 2: The relevant quotient orbifolds
  • Figure 3: Determining the homotopy class of a slope-curve
  • Figure 4: A fibered, hyperbolic knot bounding infinitely many fibered, ribbon disks that are pairwise nonisotopic modulo local knotting.
  • Figure 5: The surface $F$ is obtained by plumbing the surface $F"$ onto the surface $F'$ along the green quadrilateral, which changes the tangle $T'$ to the tangle $T$.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • ...and 32 more