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Effective bounds on characterising slopes for all knots

Patricia Sorya, Laura Wakelin

Abstract

A slope $p/q$ is characterising for a knot $K \subset \mathbb{S}^3$ if the orientation-preserving homeomorphism type of the manifold $\mathbb{S}^3_K(p/q)$ obtained by performing Dehn surgery of slope $p/q$ along $K$ uniquely determines the knot $K$. We combine new applications of results from hyperbolic geometry with previous individual work of the authors to determine, for any given knot $K$, an explicit bound $\mathcal{C}(K)$ such that $|q| > \mathcal{C}(K)$ implies that $p/q$ is a characterising slope for $K$. Furthermore, we find an optimal such $\mathcal{C}(K)$ for certain satellite knots with winding number zero patterns.

Effective bounds on characterising slopes for all knots

Abstract

A slope is characterising for a knot if the orientation-preserving homeomorphism type of the manifold obtained by performing Dehn surgery of slope along uniquely determines the knot . We combine new applications of results from hyperbolic geometry with previous individual work of the authors to determine, for any given knot , an explicit bound such that implies that is a characterising slope for . Furthermore, we find an optimal such for certain satellite knots with winding number zero patterns.

Paper Structure

This paper contains 24 sections, 25 theorems, 28 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Determining a value for C(K)
  3. Winding number zero
  4. Outline
  5. Acknowledgements
  6. Preliminaries
  7. Satellite knots
  8. JSJ decompositions
  9. Dehn surgery
  10. Hyperbolic geometry
  11. Determining a value for C(K)
  12. The hyperbolic case
  13. The cable case
  14. Proof of Theorem \ref{['theorem:C(K)']}
  15. Winding number zero
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $K \subset \mathbb{S}^3$ be a knot with exterior $\mathbb{S}^3_K$. Then the JSJ decomposition of $\mathbb{S}^3_K$ -- namely, the geometry of the JSJ pieces of $\mathbb{S}^3_K$, together with the gluing maps between them -- explicitly determines a constant $\mathcal{C}(K)$ such that if $|q| > \ma

Figures (2)

  • Figure 1: The knot $B_{-5,2}(W_{-7}(3_1), 4_1 \# 6_1)$ from Example \ref{['example:twisting']}.
  • Figure 2: A knot with "small" systole and "large" signature.

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • proof
  • Proposition 1.7
  • Theorem 1.8
  • Proposition 1.9
  • ...and 53 more