Thin knots and the cabling conjecture

DeYeso III, Robert

doi:10.2140/agt.2025.25.4547

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ISSN (electronic): 1472-2739

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Thin knots and the cabling conjecture

Robert DeYeso III

Algebraic & Geometric Topology 25 (2025) 4547–4583

DOI: 10.2140/agt.2025.25.4547

Abstract

The cabling conjecture of González-Acuña and Short states that only cable knots admit Dehn surgery to a manifold containing an essential sphere. We approach this conjecture for thin knots using Heegaard Floer homology, primarily via immersed curves techniques inspired by Hanselman’s work on the cosmetic surgery conjecture. We show that almost all thin knots satisfy the cabling conjecture, with a possible exception coming from a (conjecturally nonexistent) collection of thin, hyperbolic, $L$ -space knots. This result serves as a reproof that the cabling conjecture is satisfied by alternating knots.

Keywords

low-dimensional topology, cabling conjecture, reducible, surgery, Dehn surgery, thin knots, Heegaard Floer homology, knot Floer homology, immersed curves

Mathematical Subject Classification

Primary: 57K18

References

Publication

Received: 26 April 2022

Revised: 21 July 2024

Accepted: 27 November 2024

Published: 20 November 2025

Authors

Robert DeYeso III
Department of Mathematics & Statistics The University of Tennessee at Martin Martin, TN United States

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Volume 25, issue 8 (2025)