Kirby Problem 1.51

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.

A smooth knot $K\subset S^3=\partial B^4$ is ribbon if it bounds a smoothly embedded disk in $B^4$ whose radial Morse function has no local maxima. A symmetric-union diagram is constructed from a knot diagram and its reflected mirror: join them along the reflection axis to obtain the symmetric connected-sum diagram, and allow twists at crossings on that axis. Off the axis the diagram retains reflection symmetry. A knot is a symmetric union if it has a diagram obtained by this construction.

Statement recorded in the supplied source.

Does every smooth ribbon knot in $S^3$ admit a symmetric-union diagram? [S2]

Short English statement

Does every ribbon knot have a symmetric-union presentation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2710 (KP-1.51). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 1.51 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.