Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A homological generalized Property R conjecture is false

Tye Lidman, Trevor Oliveira-Smith, Alexander Zupan

Abstract

The generalized Property R conjecture (GPRC) predicts that if framed surgery on an $n$-component link $L$ in $S^3$ produces $\#^{n} (S^1\times S^2)$, then $L$ is handleslide equivalent to an unlink, the obvious way to construct such a surgery. Many potential counterexamples to the GPRC are known, but obstructing handleslide equivalence is a tricky proposition. In this vein, we disprove a further generalization of the GPRC. It would be reasonable to expect that if an $n$-component link in $S^3$ surgers to the connected sum of $n$ three-manifolds with the homology of $S^1 \times S^2$, then this link should be handleslide equivalent to an $n$-component split link, the obvious way to construct such a surgery. However, we prove that there are 2-component framed links in $S^3$ that surger to a connected sum of homology $S^1\times S^2$'s but that are not handleslide equivalent, or even weakly handleslide equivalent, to a split link.

A homological generalized Property R conjecture is false

Abstract

The generalized Property R conjecture (GPRC) predicts that if framed surgery on an -component link in produces , then is handleslide equivalent to an unlink, the obvious way to construct such a surgery. Many potential counterexamples to the GPRC are known, but obstructing handleslide equivalence is a tricky proposition. In this vein, we disprove a further generalization of the GPRC. It would be reasonable to expect that if an -component link in surgers to the connected sum of three-manifolds with the homology of , then this link should be handleslide equivalent to an -component split link, the obvious way to construct such a surgery. However, we prove that there are 2-component framed links in that surger to a connected sum of homology 's but that are not handleslide equivalent, or even weakly handleslide equivalent, to a split link.
Paper Structure (4 sections, 9 theorems, 6 equations, 10 figures)

This paper contains 4 sections, 9 theorems, 6 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Theorem
  4. Another perspective on the links $L_n$

Key Result

Theorem 1.1

Let $K$ be a knot in $S^3$. If Dehn surgery on $K$ yields $S^1 \times S^2$, then $K$ is unknotted.

Figures (10)

  • Figure 1: Surgery description for the Seifert fibered space $(S^2; \frac{p_1}{q_1},\ldots, \frac{p_k}{q_k})$
  • Figure 2: A $-(p,p-1)$-torus knot (left) and $-(q,2)$-torus knot (right) in $S^3$
  • Figure 3: Moves of type (1) at left, type (2) at center, and type (3) at right.
  • Figure 4: The knot $K_n$ in $S^3$, unlabeled and shown in green.
  • Figure 5: Isotopic surgery diagrams for $M_n$.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Theorem 1.1: Ga
  • Conjecture 1.2: Problem $1.82$ kirby-web, generalized Property R conjecture (GPRC)
  • Theorem 1.3
  • Conjecture 1.4: GST, weak GPRC
  • Theorem 1.5
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 9 more