Commensurability Classes of Hyperbolic Knot Complements

Definitions and mathematical statement

The supplied record does not specify the ambient manifold, the equivalence relation on knots, or orientation conventions. The following formulation is therefore conditional on fixing those conventions; these choices must be confirmed before interpreting cardinalities.

Let $\mathcal K$ be the resulting collection of knot classes, and let $M_J$ denote the complement of a representative of $J\in\mathcal K$ in its ambient manifold. Call $J$ hyperbolic if $M_J$ admits a complete finite-volume hyperbolic metric. Two complements $M_J$ and $M_K$ are commensurable if there is a connected manifold $N$ and finite-sheeted covering maps \[ N\longrightarrow M_J, \qquad N\longrightarrow M_K. \] Any requirement that these covering maps preserve orientations must be fixed separately, since the supplied record does not state one. Define \[ C(K)=\left\{J\in\mathcal K: M_J\text{ and }M_K\text{ are commensurable}\right\}. \] The problem asks, for every hyperbolic knot $K$, to describe $C(K)$, obtain upper bounds for $\lvert C(K)\rvert$, and give necessary and sufficient conditions for \[ \lvert C(K)\rvert>1. \] The record does not specify whether the requested upper bounds should be uniform over all $K$ or depend on $K$, and it does not itself assert finiteness. In particular, an interpretation using knots in $S^3$ up to ambient isotopy requires confirmation rather than being an explicit supplied hypothesis.

Short English statement

For each hyperbolic knot, describe the knot classes whose complements share a finite-sheeted cover with its complement. Seek cardinality bounds and a criterion for the class to contain more than one knot. The ambient manifold, knot-equivalence convention, and orientation requirements remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30001544$, titled Commensurability Classes of Hyperbolic Knot Complements, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The supplied proposed year is 2010; no individual proposer is identified.
• Supplied report citation and literature-source label "original OWR report": Low-Dimensional Topology and Number Theory (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/35, pp. 2101--2163. https://doi.org/10.4171/owr/2010/35. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source label "Cochran--Harvey--Leidy work": https://msp.org/agt/2008/8-3/agt-v8-n3-p15-s.pdf. The record supplies no article title or explanation establishing its relevance to this commensurability question.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says the extracted statement describes the determination of $C(K)$ as difficult and widely open, and claims that later concordance invariants and filtrations provide partial information but no classification. These are supplied dated curation claims, not independently verified conclusions. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and explains that the knot, commensurability set, and cardinality inequality were put into standard notation without changing the question. This is supplied verification metadata, not independent mathematical or source verification.