Kirby Problem 1.11

Definitions and mathematical statement

For a smooth knot $K\subset S^3$, use the meridian--preferred-longitude basis $(\mu,\lambda)$ of its exterior boundary. If $r=a/b\in\mathbb Q$ is in lowest terms, $S^3_r(K)$ denotes Dehn filling along $a\mu+b\lambda$. A closed oriented $3$-manifold is prime if any expression as a connected sum has an $S^3$ summand. Thus nonprime means a connected sum of two manifolds different from $S^3$.

A cable of a knot $J$ is a knot lying on $\partial\nu J$ as a primitive curve whose absolute longitudinal winding is at least two, pushed slightly into the solid torus. The parallel copy on that torus gives the cabling annulus in the exterior of the cable. Its boundary curves specify the cabling slope on $\partial\nu K$.

The Cabling Conjecture asserts: if $S^3_r(K)$ is nonprime, then $K$ is a nontrivial cable of some knot $J$, and $r$ is exactly its cabling slope. Describing the slope by the annulus makes the assertion independent of competing conventions for ordering the two integer cable parameters.

Short English statement

Does every nonprime surgery on a knot come from a cable knot filled along its cabling annulus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2670 (KP-1.11), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.11. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.