OpenFrontiers
Join
← Back to the index
OP-12027OpenGeometry and topology

Tight knot constant (C22aC_{22a})

Tight knot constant: determine C22aC_{22a} or improve its bounds; the repository records 1.105 (lower) and 10.76 (10.02*) (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 ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C22aC_{22a} is the largest constant for which one has an inequality

L≥C22aC3/4L\geq C_{22a}C^{3/4}

for all knots, 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.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

  • There is also a convention where the ropelength is defined relative to a unit-diameter rope, but unit-radius is more common.
  • Conjectured lower bound of 3.22 [Klotz2025] based on Mobius energy argument.
  • For knots without asymptotically large crossing number CC (currently below 1850) a stronger bound exists [Diao2003].
  • Alternating knots have a linear lower bound, discussed in 22b.

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).
  • [SDKP1998] Stasiak, Andrzej; Dubochet, Jacques; Katrich, Vsevolod; Pieranski, Piotr. Ideal knots and their relation to the physics of real knots. Ideal knots 19, 1-19 (1998).
  • [ACPR2011] Ashton, Ted; Cantarella, Jason; Piatek, Michael; Rawdon, Eric. Knot tightening by constrained gradient descent. Exp. Math. 20, No. 1, 57-90 (2011).
  • [KM2021] Klotz, Alexander R.; Maldonado, Matthew. The ropelength of complex knots. J. Phys. A 54, No. 44, 445201 (2021).
  • [DE1998] Diao, Yuanan; Ernst, Claus. The complexity of lattice knots. Topol. Appl. 90, No. 1-3, 1-9 (1998).
  • [BS1999] Buck, Gregory; Simon, Jonathan. Thickness and crossing number of knots. Topol. Appl. 91, No. 3, 245-257 (1999).
  • [Klotz2025] Klotz, Alexander. Geometric considerations for energy minimization of topological links and chainmail networks. arXiv:2507.20903
  • [Diao2003] Diao, Yuanan. The lower bounds of the lengths of thick knots. J. Knot Theory Ramifications 12, No. 01, 1-16 (2003).
  • [Klotz2026] Klotz, Alexander R. Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links. arXiv:2603.02416.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.105; best known upper bound 10.76 (10.02*). Values marked with an asterisk have minimal verification.

Known upper bounds

Upper bounds are typically found by finding a tight instance of a specific knot using gradient descent (usually a torus knot).

Bound Reference Comments
12.81 [SDKP1998] 8198_{19} knot
12.63 [ACPR2011] 1012410_{124} knot
10.76 [KM2021] T(25,26)T(25,26) knot
10.02 [Klotz2026] T(24,19)T(24,19) knot

Known lower bounds

Bound Reference Comments
0.418 [DE1998] Based on lattice embeddings.
(4π11)3/4≈1.105\left(\frac{4\pi}{11}\right)^{3/4}\approx 1.105 [BS1999] Argument based on an "electromagnetic" knot energy. The 11 in the denominator can be replaced by 10.67, bringing the bound to 1.13, but it is always reported as 11.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Tight knot constant ($C_{22a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.