Characterising slopes for hyperbolic knots and Whitehead doubles
Abstract
A slope $p/q \in \mathbb{Q}$ is characterising for a knot $K \subset \mathbb{S}^3$ if the oriented homeomorphism type of the manifold $\mathbb{S}^3_K(p/q)$ obtained by Dehn surgery of slope $p/q$ on $K$ uniquely determines the knot $K$. We combine analysis of JSJ decompositions with techniques involving lengths of shortest geodesics to find explicit conditions for a slope to be characterising for $K$ in the case where $K$ is any hyperbolic knot or any satellite knot by a hyperbolic pattern. Assuming that the list of 2-cusped orientable hyperbolic 3-manifolds obtained using the computer programme SnapPy is complete up to a certain point, we use hyperbolic volume inequalities to generate a refinement for the special case of Whitehead doubles. We also construct pairs of multiclasped Whitehead doubles of double twist knots for which $1/q$ is a non-characterising slope.
Fields of Study & Subjects
Geometric and Algebraic Topology
Botulinum Toxin and Related Neurological Disorders
Publication details
- Journal / source
- Algebraic & Geometric Topology
- Publisher
- Mathematical Sciences Publishers
- Volume
- 26
- Issue
- 2
- Pages
- 625–657
- Publication date
- 11 February 2026
- Language
- EN