Kirby Problem 1.79

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 knot diagram is a finite generic planar projection with over/under information at its crossings. Write $c(D)$ for its number of crossings. The three Reidemeister moves preserve knot type. An additional allowed diagram move must also preserve knot type. The source permits a collection of moves depending on $c(D)$, but requires that crossing-reducing instances be findable in time polynomial in $c(D)$; an arbitrary entire Reidemeister sequence declared to be one move does not meet this requirement.

Statement recorded in the supplied source.

Can one augment Reidemeister moves by such effectively findable moves so that every unknot diagram $D$ with $c(D)>0$ admits a simplification step $D\to D'$ with $c(D')<c(D)$, and repetition reaches the zero-crossing diagram? Require strict decrease at the simplification steps and a polynomial-time procedure for finding those steps, rather than merely a process with nonincreasing crossing number. [S2]

Short English statement

Find efficiently discoverable extra diagram moves that simplify every unknot by strictly decreasing the crossing count at each simplification step.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2738 (KP-1.79). 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 1.79 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.