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The Cabling Conjecture of Gonz\'alez-Acu\~na and Short holds that only cableknots admit Dehn surgery to a manifold containing an essential sphere. Weapproach this conjecture for thin knots using Heegaard Floer homology,primarily via immersed curves techniques inspired by Hanselman's work on theCosmetic Surgery Conjecture. We show that almost all thin knots satisfy theCabling Conjecture, with possible exception coming from a (conjecturallynon-existent) collection of thin, hyperbolic, L-space knots. This result servesas a reproof that the Cabling Conjecture is satisfied by alternating knots, andalso a new proof that thin, slice knots satisfy the Cabling Conjecture.

Thin knots and the Cabling Conjecture