Sharp Bounds for Exceptional Dehn-Filling Slopes

Definitions and mathematical statement

A hyperbolic knot manifold is understood here as a compact connected orientable three-manifold $M$ with torus boundary whose interior has a complete finite-volume hyperbolic metric. It is not restricted to knot complements in $S^3$. A slope is an unoriented isotopy class of essential simple closed curves on $\partial M$. The Dehn filling $M(\alpha)$ attaches a solid torus with meridian identified to slope $\alpha$.

Let $E(M)$ consist of the slopes for which $M(\alpha)$ is not hyperbolic. For slopes $\alpha,\beta$, their distance $\Delta(\alpha,\beta)$ is their minimal geometric intersection number, equal to $|ps-qr|$ for primitive coordinate pairs $(p,q),(r,s)$ in a common homology basis. Put \[ \Delta(E(M))=\max_{\alpha,\beta\in E(M)}\Delta(\alpha,\beta), \] with value zero for fewer than two slopes.

Retain both levels of the source's proposed bounds: \[ \forall M,\qquad |E(M)|\leq10,\qquad\Delta(E(M))\leq8; \] and \[ M\notin\{M_1,M_2,M_3,M_4\}\quad\Longrightarrow\quad |E(M)|\leq7,\qquad\Delta(E(M))\leq5. \] The supplied assessment reports that Lackenby--Meyerhoff proved the first pair; they are preserved as historically posed statements with that solved-part qualification. The sharper exclusion-dependent pair remains the additional target in the supplied record.

The four named exceptional manifold types $M_1,\ldots,M_4$ are not identified by constructions, surgery descriptions or census names. Their intended types must be recovered before the stronger statement has a fully specified exclusion. No particular four manifolds or restriction to $S^3$ knot complements is guessed.

Short English statement

Retain the universal exceptional-filling bounds $10$ and $8$, reported proved by Lackenby--Meyerhoff, and the sharper proposed bounds $7$ and $5$ outside four named manifold types. The four exceptions are not defined in the extract.

Sources

• Supplied report: Topologie (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/43, pp. 2419--2476; proposed year 2008. https://doi.org/10.4171/owr/2008/43.
• Supplied Lackenby--Meyerhoff reference: https://arxiv.org/abs/0808.1176.
• Supplied Gordon retrospective: https://celebratio.org/Gordon_C/article/1165/.
• Local UnsolvedMath registry, permanent ID 30001051, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note reports comparison with the official report. Literature checked at 2026-08-21: the assessment is supplied dated curation, not independently verified. It labels the record partially solved, reports proof of $|E(M)|\leq10$ and diameter at most $8$, and says the sharper bounds outside the four exceptions remain partly unresolved.