Kirby Problem 1.77

Definitions and mathematical statement

Consider diagrams of knots in the plane, modulo planar isotopy. The three Reidemeister moves change a curl, a canceling two-crossing pair, or the order of three strands around a crossing. Let $d_R(D,E)$ be the smallest number of such moves connecting diagrams $D,E$ of the same knot, assigning zero cost to planar isotopy. Intermediate diagrams may have arbitrarily many crossings. Write $c(D)$ for the number of crossings of $D$.

Determine sharp quantitative bounds, in terms of endpoint crossing numbers, for $d_R(D,E)$. One precise worst-case formulation is \[ R(n)=\max\{d_R(D,E):c(D)+c(E)\leq n,\ D,E\text{ represent the same knot}\}. \] The maximum uses combinatorial diagram types, of which there are finitely many with bounded crossing number. Reidemeister's theorem makes each distance finite. The problem asks for the growth of $R(n)$, or comparably sharp two-variable bounds in $c(D),c(E)$, valid uniformly over knot types. A particularly strong possibility is a polynomial upper bound. The quantity measures the number of moves, rather than the greatest crossing number encountered during them.

Short English statement

How many Reidemeister moves are needed in the worst case to connect two diagrams of the same knot?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2736 (KP-1.77), 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 1.77. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.