The exact minimum ropelength of a trefoil
For a closed embedded $C^{1,1}$ curve $\gamma$ in $\mathbb{R}^3$, let its thickness $\tau(\gamma)$ be its reach: the supremum of the $r > 0$ such that every point at distance less than $r$ from $\gamma$ has a unique nearest point on $\gamma$. This is the radius, not the diameter, of the thickest embedded tube around $\gamma$. The ropelength of $\gamma$ is $\mathcal{R}(\gamma) = \operatorname{Length}(\gamma)/\tau(\gamma)$, and $$\mathcal{R}_{3_1} = \inf\{\mathcal{R}(\gamma) : \gamma \text{ is ambient isotopic to a trefoil}\}.$$ Cantarella, Kusner and Sullivan proved that every knot type has a ropelength minimizer, and that minimizers are $C^{1,1}$ (2002). Denne, Diao and Sullivan proved that every nontrivial knot has ropelength at least $31.32$ in this normalization (they measure thickness as a diameter and state $15.66$; 2006). They also note that numerical experiments have found a trefoil of ropelength less than $32.744$.
Problem. Determine the exact value of $\mathcal{R}_{3_1}$.
Source: Stated by AIM — Open Applied Problems, after Cantarella, Kusner and Sullivan (2002) and Denne (2018), §7.4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- GT — Geometric Topology
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, ideal knots, ropelength
References
@article{cantarella2002,
author = {Cantarella, Jason and Kusner, Robert B. and Sullivan, John M.},
title = {On the minimum ropelength of knots and links},
journal = {Invent. Math.},
volume = {150},
number = {2},
year = {2002},
pages = {257--286},
doi = {10.1007/s00222-002-0234-y},
eprint = {math/0103224},
archivePrefix = {arXiv},
}
@incollection{denne2018,
author = {Denne, Elizabeth},
title = {Quadrisecants and essential secants of knots},
booktitle = {New Directions in Geometric and Applied Knot Theory},
publisher = {De Gruyter},
address = {Berlin},
year = {2018},
pages = {138--158},
doi = {10.1515/9783110571493-007},
eprint = {1608.02608},
archivePrefix = {arXiv},
}
@article{denne2006,
author = {Denne, Elizabeth and Diao, Yuanan and Sullivan, John M.},
title = {Quadrisecants give new lower bounds for the ropelength of a knot},
journal = {Geom. Topol.},
volume = {10},
year = {2006},
pages = {1--26},
doi = {10.2140/gt.2006.10.1},
eprint = {math/0408026},
archivePrefix = {arXiv},
}
@misc{aim-ideal-trefoil-ropelength,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The exact minimum ropelength of a trefoil}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/237-ideal-trefoil-ropelength.md},
note = {Entry 237 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00600, "The exact minimum ropelength of a trefoil", clopen.solutions.