Thin knots and the Cabling Conjecture

TL;DR

The paper addresses the Cabling Conjecture for thin knots by leveraging Heegaard Floer homology and immersed-curve techniques to study reducible Dehn surgeries. The authors show that, for thin hyperbolic knots, a reducible surgery forces the knot to be an $L$-space knot with slope $r=2g(K)-1$ (after mirroring), implying the Cabling Conjecture holds for alternating knots and for thin, slice knots. The approach unifies bordered and immersed-curve perspectives to constrain Maslov gradings and involves a detailed case analysis based on $\tau(K)$ relative to the genus. The results significantly narrow potential counterexamples to the Cabling Conjecture among thin knots and provide a reproof of existing cases for alternating knots. The work highlights how Heegaard Floer invariants, especially $d$-invariants and the $V_s/H_s$ data, interact with the geometry of Dehn surgeries to produce strong obstructions to reducible surgeries.

Abstract

The Cabling Conjecture of González-Acuña 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.

Figures (12)

• Figure 1: Edges of a grading arrow either follow or oppose the orientations of the attached curve components.
• Figure 2: An example of $\widehat{\textit{HF}}(M)$ for $g(K)=2$ and $\tau(K)=1$.
• Figure 3: The pairing of $\widehat{\textit{HF}}(S^3 \setminus \nu T(2,5))$ and $h(\widehat{\textit{HF}}(D^2 \times S^1))$, whose intersection Floer homology is $\widehat{\textit{HF}}(S^3_4(T(2,5))$.
• Figure 4: Bigons used to determine the relative Maslov grading. Example (a) does not involve a grading arrow, while (b) and (c) (with a cusp) do.
• Figure 5: The possibilities for the reference intersection $a^s$. (a) has $\tau(K) = 0$, (b) has $\tau(K) > 0$ and $|s| < \tau(K)$, and (c) has $\tau(K) < 0$ with two curves representing $s \geq 0$ in red and $s<0$ in purple. The case when $\tau(K) > 0$ and $|s| \geq \tau(K)$ is similar to (a).

Theorems & Definitions (42)

• Conjecture 1.1: Cabling Conjecture, Gonzalez-Acuña -- Short GAS86
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Corollary 1.6
• Lemma 2.1
• proof
• Lemma 2.2: NW15
• Proposition 2.3: NW15