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OP-12028OpenGeometry and topology

Tight alternating knot constant (C22bC_{22b})

Tight alternating knot constant: determine C22bC_{22b} or improve its bounds; the repository records 0.017 (lower) and 7.31 (upper).

MSC 57K10 · Knot theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

C22b=boC_{22b} = b_{o} is the largest constant for which one has an inequality

L≥boCL \geq b_{o} C

for all knots that admit an alternating diagram, where LL is the ropelength of a knot (or link) with crossing number CC. The ropelength LL is the infimum over all embeddings of the knot (or link) of the ratio of the contour length of the knot to its thickness. The thickness is defined as the radius of the smallest circle that passes through any three points on the knot (where collinear points yield an infinite radius). Colloquially, the ropelength is the least amount of rope required to tie a specific knot in a rope of unit radius. See [CKS2002] for the full definition.

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Background

  • There is also a convention where the ropelength is defined relative to a unit-diameter rope, but unit-radius is more common.
  • Non-alternating knots have a three-quarter power lower bound, discussed in 22a.

References (from the repository page)

  • [CKS2002] Cantarella, Jason; Kusner, Robert B.; Sullivan, John M. On the minimum ropelength of knots and links. Invent. Math. 150, No. 2, 257-286 (2002).
  • [Diao2024] Diao, Yuanan. The ropelength conjecture of alternating knots. Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 177. No. 2. Cambridge University Press, 2024.
  • [Olsen2013] Olsen, Kasper W., and Jakob Bohr. A principle for ideal torus knots. Europhysics Letters 103.3 (2013): 30002.
  • [Huh2018] Huh, Youngsik, Hyoungjun Kim, and Seungsang Oh. Ropelength of superhelices and (2, n)-torus knots. Journal of Physics A: Mathematical and Theoretical 51.48 (2018): 485203.
  • [Klotz2021] Klotz, Alexander R., and Matthew Maldonado. The ropelength of complex knots. Journal of Physics A: Mathematical and Theoretical 54.44 (2021): 445201.
  • [Kim2024] Kim, Hyoungjun, Seungsang Oh, and Youngsik Huh. Efficiency of non-identical double helix patterns in minimizing ropelength of torus knots. Physica Scripta 99.7 (2024): 075240.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.017; best known upper bound 7.31. Values marked with an asterisk have minimal verification.

Known upper bounds

Upper bounds are typically found by constructing alternating torus knots or links and minimizing the parameters of their construction.

Bound Reference Comments
2π+2≈8.282\pi+2\approx 8.28 Trivial Hopf chain link of stadium curves
8.50 [Olsen2013] Double helix
7.63 [Huh2018] Four-strand superhelix
1+π4+1π2≈7.361+\pi\sqrt{4+\frac{1}{\pi^2}}\approx 7.36 [Klotz2021] Wrapped circle
7.31 [Kim2024] Asymmetric double helix

Known lower bounds

Bound Reference Comments
159.5≈0.017\frac{1}{59.5}\approx 0.017 [Diao2024] First proof of positivity
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Tight alternating knot constant ($C_{22b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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