Kirby Problem 1.54

Definitions and mathematical statement

Two oriented knots are smoothly concordant if they bound an embedded smooth annulus in $S^3\times[0,1]$. For a fixed knot $K$, let $\mathscr K_{[K]}$ be the set of oriented isotopy classes smoothly concordant to $K$. On this set put $J_0\preceq J_1$ when an embedded annulus from $J_0$ to $J_1$ has the ambient product height restricting to a Morse function with no index-two critical points. This is the ribbon-concordance partial order, directed so that the unknot is below a ribbon knot.

Determine the order structure of $\mathscr K_{[K]}$. In particular:

1. What order types can arise from concordance classes?
2. Does the order type depend on $[K]$, and does every class contain a minimal element? If a minimal element exists, must it be unique?
3. Can some class contain two distinct minimal elements?

Here $J$ is minimal if the only $J'$ in the class with $J'\preceq J$ is $J$ itself. A least element, if present, must instead lie below every member of the class; these are different order-theoretic requirements. Order type means isomorphism as a partially ordered set, without choosing knot diagrams.

Short English statement

What does the ribbon-concordance order look like inside one smooth concordance class, and can it have more than one minimal knot?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2713 (KP-1.54), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.54. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.