Kirby Problem 1.18

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

All knots and links below are smooth in the oriented $3$-sphere, up to ambient isotopy, unless another ambient manifold is specified. The meridional rank $\mu(L)$ is the least size of a generating set of $\pi_1(S^3\setminus L)$ consisting of meridians (loops linking one component once, transported to a basepoint). The bridge number $b(L)$ is the minimum number of local maxima in a bridge presentation. Two groups have the same finite quotients if the same finite groups occur, up to isomorphism, as quotients. A hyperbolic knot has a complete finite-volume hyperbolic metric on its complement.

Statement recorded in the supplied source.

(a) Is $\mu(L)=b(L)$ for every link? (b) For knots whose groups have the same finite quotients, must their meridional ranks agree; can meridional rank be explicitly recovered from these quotients? (c) Does a meridian-preserving epimorphism of link groups force $b(L_1)\geq b(L_2)$? (d) For an arbitrary epimorphism of knot groups, must both $\mu(K_1)\geq\mu(K_2)$ and $b(K_1)\geq b(K_2)$ hold? (e) For a hyperbolic knot $K$, is every subgroup generated by at most $b(K)-1$ meridians free? [S1][S2]

Short English statement

Relate meridional generators and bridges, determine what finite quotients and epimorphisms detect, and test freeness of small meridional subgroups of hyperbolic knot groups.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2677 (KP-1.18). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.18 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.