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Dehn filling and the knot group II: Ubiquity of persistent elements

Tetsuya Ito, Kimihiko Motegi, Masakazu Teragaito

Abstract

Let $K$ be a nontrivial knot in $S^3$. We say that an element of the knot group $G(K)$ is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is equivalent to the statement that the meridian of $K$ is a persistent element, and this represents the first instance of such elements. Building on the solution to the Property P conjecture due to Kronheimer and Mrowka, we show that every nontrivial knot group admits infinitely many persistent elements with pairwise disjoint automorphic orbits, none of which contains a power of the meridian. We then develop this further to show that for a broad class of hyperbolic knots - namely those admitting no surgery whose resulting manifold has torsion in its fundamental group - persistent elements are not rare curiosities, but rather structurally pervasive in $G(K)$. This is reflected in the following two properties: (i) Every subgroup of $G(K)$ that is not contained in the normal closure of a peripheral element contains persistent elements. (ii) Persistent elements exist outside every proper subgroup of $G(K)$.

Dehn filling and the knot group II: Ubiquity of persistent elements

Abstract

Let be a nontrivial knot in . We say that an element of the knot group is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is equivalent to the statement that the meridian of is a persistent element, and this represents the first instance of such elements. Building on the solution to the Property P conjecture due to Kronheimer and Mrowka, we show that every nontrivial knot group admits infinitely many persistent elements with pairwise disjoint automorphic orbits, none of which contains a power of the meridian. We then develop this further to show that for a broad class of hyperbolic knots - namely those admitting no surgery whose resulting manifold has torsion in its fundamental group - persistent elements are not rare curiosities, but rather structurally pervasive in . This is reflected in the following two properties: (i) Every subgroup of that is not contained in the normal closure of a peripheral element contains persistent elements. (ii) Persistent elements exist outside every proper subgroup of .

Paper Structure

This paper contains 17 sections, 19 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation
  4. Results
  5. Persistent elements and pseudo-meridians
  6. Persistent elements in a knot group
  7. Baumslag-Solitar relators
  8. Proof of Theorem \ref{['existence_persistent_elements']}
  9. Minimizers for subgroups
  10. Existence of minimizers
  11. Shrinking Lemma
  12. Characterization of subgroups containing persistent elements
  13. Complements of subgroups and persistent elements
  14. Pseudo-meridians and persistent elements
  15. Further questions
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any nontrivial knot, its meridian $\mu$ survives under all nontrivial Dehn fillings, i.e. $\mathcal{S}(\mu) = \emptyset$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 6.1: Longitudes $\lambda_1, \lambda_2$, and $\lambda = \lambda_1 \lambda_2$

Theorems & Definitions (53)

  • Theorem 1.1: Reformulation of Property P KM
  • Theorem 1.3: non-meridional persistent elements
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6: Existence of minimizer
  • Theorem 1.7
  • Corollary 1.8
  • Example 1.9: Seifert surface subgroup
  • Remark 1.10
  • Corollary 1.11: Finite index subgroup
  • ...and 43 more