Kirby Problem 1.38

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.

The smooth concordance group $\mathcal C$ consists of oriented knots modulo smooth annuli in $S^3\times[0,1]$, with connected sum as addition and unknot $U$ as zero. A knot is smoothly slice if it bounds a smooth disk in $B^4$, and topologically slice if it bounds a locally flat disk. For a pattern $P\subset S^1\times D^2$, the satellite $P(K)$ uses the preferred longitude; its winding number is the signed homology class in the solid torus. Satellite operations induce functions on $\mathcal C$, not necessarily homomorphisms. Write $\mathcal C_{TS}\leq\mathcal C$ for the subgroup of classes of topologically slice knots. The order of a class is the least positive integer $m$ with $m[K]=0$, if one exists; a nontrivial torsion class has finite order greater than one. A knot $J$ is strongly negatively amphichiral if there is an orientation-reversing smooth involution of $S^3$ preserving $J$ setwise and reversing its orientation. The divisibility condition in part(c) is exactly division by infinitely many positive integers, not division by every integer.

Statement recorded in the supplied source.

Determine the structure of $\mathcal C$, with the following specific questions. (a) Does $\mathcal C$ contain a nontrivial torsion class of order other than two, and does $\mathcal C_{TS}$ contain such a class? (b) Does some class of order two in $\mathcal C$ have no strongly negatively amphichiral representative? (c) Does there exist a knot $K$ such that the set $\{n\in\mathbb N:\exists[J]\in\mathcal C,\ [K]=n[J]\}$ is infinite? Part(c) is recorded literally: no nonzero or infinite-order hypothesis is imposed on $[K]$; in particular the unknot satisfies this condition. Likewise a finite-order class is divisible by every positive $n$ coprime to its order. [S1][S2]

Short English statement

Study the smooth concordance group and its topologically slice subgroup, asking about higher torsion, representatives of order-two classes and the stated infinitely-many-integers divisibility condition.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2697 (KP-1.38). 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 1.38 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.