Minimum-Volume Hyperbolic Knots in the Three-Torus

Definitions and mathematical statement

Let $T^3=S^1\times S^1\times S^1$, and let $v_8$ denote the hyperbolic volume of a regular ideal octahedron in hyperbolic three-space of sectional curvature $-1$. Let $B\subset S^3$ be the Borromean rings.

The supplied record does not define topological volume or specify the class of admissible links. For an explicit, conditional link-based formulation, let $\mathcal H(M)$ consist of nonempty finite tame links $L\subset M$ such that $M\setminus L$ admits a complete finite-volume hyperbolic metric of sectional curvature $-1$, and set \[ \operatorname{vol}_t(M)=\inf_{L\in\mathcal H(M)}\operatorname{Vol}(M\setminus L). \] Whether this is precisely the source's intended definition, including whether knots alone are admissible, requires verification.

Under this interpretation, the supplied conjecture asks whether \[ \operatorname{vol}_t(T^3)=2v_8, \] with the infimum attained by a three-component link $L_0\subset T^3$ whose complement is homeomorphic to $S^3\setminus B$. Equivalently, the requested assertion is that there exists such an $L_0\in\mathcal H(T^3)$ satisfying \[ T^3\setminus L_0\cong S^3\setminus B, \qquad \operatorname{Vol}(T^3\setminus L_0)=2v_8, \] and that every $L\in\mathcal H(T^3)$ satisfies \[ \operatorname{Vol}(T^3\setminus L)\geq 2v_8. \] The supplied excerpt presents the candidate upper bound $\operatorname{vol}_t(T^3)\leq\operatorname{Vol}(S^3\setminus B)=2v_8$; this draft records that assertion without independently verifying its construction.

The excerpt also discusses an undefined knot $K'$, an unspecified fibration, a punctured sphere, and a guts argument attributed to an unidentified reference [2]. It states that failure of $K'$ to be isotoped transverse to the fibration gives exterior volume at least $v_8=3.66\ldots$, and concludes that it is not a volume minimizer. It further suggests using a requirement that a minimizer be a braid with at least five strands to identify it with $\mathrm{m043}$. The hypotheses and ambient manifold for these statements are absent. They cannot be incorporated as constraints on the three-torus conjecture without additional source context.

Short English statement

Under the explicitly stated link-based interpretation of topological volume, the conjecture asks whether every finite-volume hyperbolic link complement in $T^3$ has volume at least $2v_8$, with equality attained by a three-component link having the Borromean-rings complement. The source definition and the relevance of the adjacent knot discussion remain unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $30006055$, source problem code catalogue entry, title "Minimum-Volume Hyperbolic Knots in the Three-Torus." The supplied record is the evidence used for this draft; no individual proposer is supplied.
• Supplied source citation: MATRIX-MFO Tandem Workshop: Invariants in Low-Dimensional Topology: Combinatorics, Geometry, and Computation (2025), Oberwolfach Reports, pp. 2511--2550, DOI 10.4171/owr/2024/43. https://doi.org/10.4171/owr/2024/43. The supplied literature-source label is "Oberwolfach Report 43/2024."
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem open and reports that the source provides a candidate and structural constraints but leaves equality with the candidate volume conjectural, without completing a finite-census or theoretical argument. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked and its notation repaired. These are supplied provenance claims; this draft does not claim to have visited or independently verified the report.