Effective bounds on characterising slopes for all knots

TL;DR

This paper establishes explicit bounds on slopes that uniquely determine knots via Dehn surgery, combining hyperbolic geometry and previous work to identify characterising slopes for all knots.

Contribution

It provides explicit bounds for characterising slopes for any knot, improving understanding of knot uniqueness in Dehn surgery outcomes.

Findings

Explicit bounds (K) for characterising slopes are determined.

Optimal bounds are found for certain satellite knots.

The approach combines hyperbolic geometry with previous knot theory results.

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.