Characterising slopes for hyperbolic knots and Whitehead doubles

Wakelin, Laura

doi:10.2140/agt.2026.26.625

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Characterising slopes for hyperbolic knots and Whitehead doubles

Laura Wakelin

Algebraic & Geometric Topology 26 (2026) 625–657

DOI: 10.2140/agt.2026.26.625

Abstract

A slope $p ∕ q \in ℚ$ is characterising for a knot $K \subset 𝕊^{3}$ if the oriented homeomorphism type of the manifold $𝕊_{K}^{3} (p ∕ q)$ 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 noncharacterising slope.

Keywords

Dehn surgery, characterising slopes, hyperbolic geometry

Mathematical Subject Classification

Primary: 57K10, 57K32

References

Publication

Received: 28 August 2024

Revised: 20 January 2025

Accepted: 9 February 2025

Published: 11 February 2026

Authors

Laura Wakelin
Department of Mathematics King’s College London London United Kingdom

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Volume 26, issue 2 (2026)