Kirby Problem 1.40

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. The Whitehead doubling operator is the standard untwisted Whitehead pattern, of winding zero. Write $P(nK)$ for the satellite of the $n$-fold concordance sum, allowing negative $n$. The rank of an abelian group $H$ is $\dim_{\mathbb Q}(H\otimes\mathbb Q)$, and $\langle X\rangle$ means the subgroup generated by $X$.

Statement recorded in the supplied source.

(a) If $w(P)=1$, $P(U)$ is concordant to $U$, and $P(K)$ is concordant to $U$, must $K$ be concordant to $U$? (b) Is untwisted Whitehead doubling injective on $\mathcal C$, and is any winding-zero satellite operator injective? (c) Are the only homomorphisms induced by satellite patterns the zero map, identity, and orientation-reversal involution? (d) For every nonconstant winding-zero satellite function, does $\langle\operatorname{im}P\rangle$ have infinite rank? More specifically, must there exist $K$ with $\langle P(nK):n\in\mathbb Z\rangle$ of infinite rank, and must this hold for every $K$ whose multiples form a rank-one subgroup? [S1][S2]

Short English statement

Test injectivity of satellite operators, classify those that are homomorphisms, and ask how large the image of every nonconstant winding-zero operator must be.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2699 (KP-1.40). 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 1.40 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.