Thin knots and the Cabling Conjecture

TL;DR

This paper proves that most thin knots satisfy the Cabling Conjecture using Heegaard Floer homology, with potential exceptions only among hypothetical thin, hyperbolic, L-space knots, providing new insights into knot theory.

Contribution

It demonstrates that almost all thin knots adhere to the Cabling Conjecture, offering a new proof for alternating and thin, slice knots.

Findings

Almost all thin knots satisfy the Cabling Conjecture.

The result reaffirms the conjecture for alternating knots.

Provides a new proof for thin, slice knots satisfying the conjecture.

Abstract

The Cabling Conjecture of Gonz\'alez-Acu\~na and Short holds 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 possible exception coming from a (conjecturally non-existent) collection of thin, hyperbolic, L-space knots. This result serves as a reproof that the Cabling Conjecture is satisfied by alternating knots, and also a new proof that thin, slice knots satisfy the Cabling Conjecture.