Is the upper bound of a closed knot's minimum ropelength linear to its crossing number

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A knot type $K$ is an ambient-isotopy class of embedded closed curves in $\mathbb R^3$. For a $C^{1,1}$ representative $\gamma$ (continuously differentiable with Lipschitz derivative), its thickness $\tau(\gamma)$ is the supremal radius of disjoint normal disks forming an embedded tube. Its ropelength is $\operatorname{length}(\gamma)/\tau(\gamma)$, and $L(K)$ is the infimum over all representatives. This is the radius convention for unit thickness. The crossing number $\operatorname{Cr}(K)$ is the minimum number of transverse crossings in a regular planar knot diagram.

Statement recorded in the supplied source.

Does there exist a constant $C>0$, independent of the knot type, such that $L(K)\leq C\max\{1,\operatorname{Cr}(K)\}$ for every knot type $K$? Equivalently, is $L(K)=O(\operatorname{Cr}(K))$ uniformly over nontrivial knot types? [S2]

Short English statement

Can every nontrivial knot be tied with unit-radius rope whose length is bounded by a universal constant times its crossing number?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200083 (AMR-071-0083). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Yuanan Diao, Claus Ernst, Attila Por and Uta Ziegler, The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers (2009), v1; arXiv:0912.3282. https://arxiv.org/pdf/0912.3282.