Kirby Problem 1.4

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. A satellite pattern $P$ is a knot in the solid torus $V=S^1\times D^2$, with winding number $w(P)=|[P]|$ under $H_1(V;\mathbb Z)\cong\mathbb Z$. It is nontrivial if it is not isotopic in $V$ to the core. For a companion $K$, $P(K)$ is formed by identifying $V$ with a tubular neighborhood of $K$ using the preferred zero-framed longitude. The unknotting number $u(J)$ is the least number of crossing changes turning $J$ into the unknot.

Statement recorded in the supplied source.

For every nontrivial pattern $P$ with $w(P)>0$ and every nontrivial companion knot $K$, is $u(P(K))\geq w(P)+1$? [S1][S2][S3]

Short English statement

A nontrivial satellite pattern with nonzero winding should force more crossing changes than its winding number, for every nontrivial companion.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2663 (KP-1.4). 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 1.4 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Jennifer Hom, Tye Lidman and JungHwan Park, Unknotting number and cabling, Conjecture 1.10. https://arxiv.org/abs/2206.04196.