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Generalized square knots and homotopy 4-spheres

Jeffrey Meier, Alexander Zupan

Abstract

The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $\#^n(S^1\times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diffeomorphic to the standard 4-sphere in the special case that one of the 2-handles is attached along a knot of the form $Q_{p,q} = T_{p,q}\#T_{-p,q}$, which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf. Along the way, we use thin position techniques from Heegaard theory to give a characterization of 2R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot $Q_{p,q}$. As a consequence, we produce large families, for all even $n$, of $n$R-links that are potential counterexamples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.

Generalized square knots and homotopy 4-spheres

Abstract

The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing -component links with a Dehn surgery realizing . We call such links R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diffeomorphic to the standard 4-sphere in the special case that one of the 2-handles is attached along a knot of the form , which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf. Along the way, we use thin position techniques from Heegaard theory to give a characterization of 2R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot . As a consequence, we produce large families, for all even , of R-links that are potential counterexamples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.

Paper Structure

This paper contains 21 sections, 60 theorems, 35 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Slice knots and links
  4. Fibered, homotopy-ribbon knots
  5. Stable equivalence classes of 2R-links
  6. Curves on the fiber of a generalized square knot
  7. Fibering generalized square knots
  8. The Seifert fibered structure of $Y_Q$
  9. Lifting curves and Dehn twists from the pillowcase
  10. Classifying the CG-derivatives of $Q_{p,q}$
  11. The link $Q_{p,q} \cup \mathcal{L}_{0}$ has Property R
  12. Twisting on vertical tori in $Y_Q$
  13. Twisting the torus $W_0$
  14. Twisting the torus $W_\infty$
  15. Standarding the Casson-Gordon Spheres
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that $X$ is a homotopy 4--sphere that can be built with no 1--handles and two 2--handles such that the attaching sphere of one of the 2--handles is a generalized square knot $Q_{p,q}$. Then $X$ is diffeomorphic to $S^4$.

Figures (13)

  • Figure 1: A schematic of a Heegaard double $(Y;S,\Delta,h)$, where $Y$ is split into two compression bodies $C^\pm$ with $\Sigma^\pm = \partial_+C^\pm$ and $\partial_-C^- = \partial_-C^+ = S$.
  • Figure 2: A sequence of schematics describing the process of untelescoping a Heegaard double in the case that $S_1$ is connected.
  • Figure 3: A local picture of the Heegaard torus $T$ near where the boundaries of the disks $D_i$ and $D_j'$ intersect, where the intersection point $x_{i,j}$ has been replaced with the band $B_{i,j}$.
  • Figure 4: (Left) The surface fiber $F^+$ for the torus knot $K^+$, shown with spine graph $\Gamma^+$. (Center) The annulus $A^+$ obtained by cutting open $F^+$ along $\Gamma^+$. (Right) The (punctured) annulus $A$ whose edge identifications yield the surface fiber $F$ for $Q$ . Shown in orange is the arc $\eta_{1,1}$ on $F^+$ and the curve $V_{1,1}$ on $F$.
  • Figure 5: The local model of the monodromy $\varphi$ of $Q$ near the puncture of $A$, featuring the necessary action of dragging the puncture back to its initial position after the $1/pq$ clockwise rotation.
  • ...and 8 more figures

Theorems & Definitions (112)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 102 more