Kirby Problem 1.53

Definitions and mathematical statement

Write $K_0\preceq K_1$ if there is a smooth annulus in $S^3\times[0,1]$ from $K_0$ at height zero to $K_1$ at height one whose height function has only index-zero and index-one critical points. This fixes the direction of ribbon concordance; in particular the unknot lies below any ribbon knot. A property $\mathcal P$ is hereditary if $K_1\in\mathcal P$ and $K_0\preceq K_1$ imply $K_0\in\mathcal P$. A knot $K_1$ is minimal if $K_0\preceq K_1$ implies the knots are isotopic. A numerical invariant $I$ is monotone if $I(K_0)\leq I(K_1)$ whenever $K_0\preceq K_1$.

Determine hereditary properties, criteria for minimality, and monotone invariants. The specified test questions are:

1. Is being alternating hereditary?
2. Is every positive knot minimal?
3. Are crossing number, bridge number, and braid index monotone?

Alternating and positive mean admitting, respectively, an alternating diagram or a diagram with all crossings positive. Crossing number minimizes diagram crossings; bridge number minimizes maxima of a height function over embeddings; braid index minimizes the number of strands among braid closures representing the knot.

Short English statement

Which features become simpler when one moves downward in the ribbon-concordance order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2712 (KP-1.53), 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 1.53. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.