Trace Fields of Hyperbolic Link Complements

Definitions and mathematical statement

A number field is a finite extension of ${\mathbb{Q}}$. Under the usual convention that a link complement means a complement in the three-sphere, let $L\subset S^3$ be a finite disjoint union of smoothly embedded circles, and suppose that $M=S^3\setminus L$ admits a complete hyperbolic metric of sectional curvature $-1$ and finite volume. Choose an orientation and a holonomy representation $\rho:\pi_1(M)\to\operatorname{PSL}_2({\mathbb{C}})$. Define the invariant trace field by \[ k(M)={\mathbb{Q}}\left(\operatorname{tr}(\widetilde{\rho(\gamma)})^2:\gamma\in\pi_1(M)\right)\subset{\mathbb{C}}, \] where $\widetilde{\rho(\gamma)}\in\operatorname{SL}_2({\mathbb{C}})$ is either lift of $\rho(\gamma)$. The squared trace is independent of the sign of the chosen lift; no compatible choice of lifts is required in this definition. We regard number fields up to ${\mathbb{Q}}$-isomorphism.

The classification problem, with these conventions made explicit, is to characterize the collection \[ \left\{[K]_{{\mathbb{Q}}}: [K:{\mathbb{Q}}]<\infty,\quad \exists L\subset S^3\ \text{such that }S^3\setminus L\text{ is complete hyperbolic of finite volume and }K\cong_{{\mathbb{Q}}}k(S^3\setminus L)\right\}. \] Equivalently, determine necessary and sufficient conditions on a number field $K$ for such a link $L$ to exist.

The supplied statement does not explicitly specify the ambient three-manifold, the completeness and finite-volume conventions, or whether fields are classified as embedded subfields of ${\mathbb{C}}$ or up to isomorphism. The formulation above uses the standard sphere-complement and finite-volume interpretation, and classification up to ${\mathbb{Q}}$-isomorphism; these conventions require confirmation against the original source.

Short English statement

Determine exactly which number fields occur as invariant trace fields of hyperbolic link complements. The displayed formulation uses complete finite-volume complements of links in $S^3$ and identifies fields up to ${\mathbb{Q}}$-isomorphism; the supplied wording leaves these conventions implicit.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004117$, source problem code catalogue entry, title Trace Fields of Hyperbolic Link Complements. The supplied record labels the statement exact and the problem open. These are supplied registry labels, not independent verification.
• Supplied report citation: Mini-Workshop: Reflection Groups in Negative Curvature (2020), Oberwolfach Reports, pp. 1043--1070, DOI 10.4171/owr/2019/17. https://doi.org/10.4171/owr/2019/17. The background attributes the collection to workshop open-problem discussions. The record gives proposed year $2019$ and names no individual proposer; the year $2020$ in the report title is retained as supplied.
• Supplied literature source: Invariant trace fields of link complements. https://arxiv.org/abs/1811.00679.
• Supplied literature source: Oberwolfach reflection-group report. https://doi.org/10.4171/owr/2019/17.
• Literature assessment dated exactly 2026-08-22: the supplied curation states that infinitely many CM fields are realized by explicit families of link complements, while the general classification remains open. Here a CM field means a totally imaginary quadratic extension of a totally real number field. This assessment is supplied dated curation derived from the OWR collection, not independently verified; the record expressly requests independent verification of time-sensitive status and cited claims.