Kirby Problem 1.71

Definitions and mathematical statement

For a closed $C^{1,1}$ embedded curve $C\subset\mathbb R^3$, its thickness is its reach, \[ \tau(C)=\sup\{r>0:\text{the nearest-point projection }U_r(C)\to C \text{ is single-valued}\}. \] Here $U_r(C)=\{x:\operatorname{dist}(x,C)<r\}$. Fix a length $\ell\geq2\pi$. A physical unknot is an unknotted curve of length $\ell$ and thickness at least one. A physical isotopy is a continuous path of embedded curves of that same length, all satisfying the same thickness bound. Allow reparametrization of the curves; the constraints concern their images.

Does there exist a physical unknot $C$ for which no physical isotopy reaches a round circle of length $\ell$? Such an example is called a Gordian unknot. More generally, two ordinary isotopic curves of the same fixed length that lie in distinct components of this constrained configuration space form a Gordian pair.

The round comparison circle has radius $\ell/(2\pi)\geq1$, so it is itself an admissible configuration. The obstruction sought must come from the global length-and-thickness constraint, even though an ambient isotopy without those metric restrictions unknots the curve.

Short English statement

Can a thick unknotted loop be trapped away from a round circle when its length and minimum thickness must be preserved?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2730 (KP-1.71), 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 1.71. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.