Math Conjectures

Cabling conjecture

OPENMajorConjectureProposed 1983 · Standard version

Canonical statement

Let K⊂S3K\subset S^{3} be a nontrivial knot, let r∈Q∪{∞}r\in\mathbb Q\cup\{\infty\} be a slope in the meridian--longitude basis, and let Sr3(K)S^{3}_{r}(K) denote the result of rr-Dehn filling the exterior of KK. If Sr3(K)S^{3}_{r}(K) is reducible, then there are coprime integers p,qp,q with q≥2q\ge2 such that KK is obtained by placing the (p,q)(p,q)-torus-knot pattern in a tubular neighborhood of a companion knot, and r=pqr=pq.
View source LaTeX
Let \(K\subset S^{3}\) be a nontrivial knot, let \(r\in\mathbb Q\cup\{\infty\}\) be a slope in the meridian--longitude basis, and let \(S^{3}_{r}(K)\) denote the result of \(r\)-Dehn filling the exterior of \(K\). If \(S^{3}_{r}(K)\) is reducible, then there are coprime integers \(p,q\) with \(q\ge2\) such that \(K\) is obtained by placing the \((p,q)\)-torus-knot pattern in a tubular neighborhood of a companion knot, and \(r=pq\).
GEO-TOP-017:[GonzalezAcunaShort1986Cabling][Scharlemann1990ReducibleSurgery][DeyKingShawTosun2021ThinKnots]
Geometry and topology

Notes

Dehn surgery on a nontrivial knot in S3S^3 can occasionally produce a reducible 33-manifold — one containing an essential sphere — and there is one well-known mechanism: surgery on a (p,q)(p,q)-cable knot with q≥2q\ge 2 along its cabling slope r=pqr=pq always does so. The cabling conjecture, posed in 1983 by González-Acuña and Short and appearing in their paper on knot surgery and primeness [GonzalezAcunaShort1986Cabling], asserts that this is the only mechanism: if Sr3(K)S^3_r(K) is reducible, then KK is a cable knot and rr is its cabling slope.

Reducible surgeries are known to be strongly constrained, with Scharlemann's analysis of producing reducible manifolds by surgery a cornerstone of these restrictions [Scharlemann1990ReducibleSurgery]. The conjecture has also been verified for many classes of knots, including recent progress on thin knots [DeyKingShawTosun2021ThinKnots].

What is missing is a general argument excluding a reducible surgery on a knot that is not a cable; despite the accumulated constraints, the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.