Kirby Problem 1.78

Definitions and mathematical statement

Let $D$ be a diagram of the unknot and let $c(D)$ count its crossings. Among all finite sequences of Reidemeister moves from $D$ to a zero-crossing circle, define the smallest achievable maximum crossing number by \[ h(D)=\min_{D=D_0\to D_1\to\cdots\to D_m=U} \max_{0\leq i\leq m}c(D_i). \] Planar isotopy is allowed freely between moves. All intermediate diagrams represent the unknot, but their crossing numbers can increase. For each integer $n\geq0$, put \[ h(n)=\max\{h(D):D\text{ is an unknot diagram with exactly }n\text{ crossings}\}. \] Use combinatorial plane diagram types modulo planar isotopy, making the outer maximum finite. The inner minimum exists because the Reidemeister moves connect all diagrams of the unknot.

Determine $h(n)$, or its sharp growth as $n\to\infty$. Since the initial diagram is included, $h(n)\geq n$. A hard unknot diagram satisfies $h(D)>c(D)$, meaning every simplification must first pass above its original crossing count. The problem concerns this necessary crossing excess, independent of how many moves the chosen simplification uses.

Short English statement

How many extra crossings can be unavoidable while simplifying an $n$-crossing diagram of the unknot?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2737 (KP-1.78), 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 1.78. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.