Kirby Problem 3.11

Definitions and mathematical statement

Let $K\subset S^3$ be a knot whose complement $M=S^3\setminus K$ admits a complete finite-volume hyperbolic metric of curvature $-1$. It has one cusp. Expand a horospherical cusp neighborhood until its interior is maximal embedded. Its boundary carries the limiting flat torus metric of this maximal cusp; this fixes the scale for all slope lengths below.

A slope is an unoriented isotopy class of essential simple closed curves on the peripheral torus. The meridian $\mu$ is the slope bounding a disk in the removed tubular neighborhood of $K$. In the maximal-cusp metric its geodesic representative has a well-defined Euclidean length $\ell_M(\mu)$, equivalently the length of its primitive vector in the cusp lattice.

Is the universal inequality \[ \ell_M(\mu)\leq4 \] valid for every hyperbolic knot in $S^3$? The upper bound is measured on the maximal cusp, not on an arbitrarily rescaled horotorus. The ambient sphere and the one-component knot condition are part of the assertion: analogous claims for multicomponent links or knots in other manifolds are different questions. A proof or a single knot with meridian length greater than four would decide the proposed bound.

Short English statement

Is the meridian of every hyperbolic knot in the three-sphere at most four units long on its maximal cusp torus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2809 (KP-3.11), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.11. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.