What constant $1.1 < a \leq 10.76$ governs the lower bound of a closed knot $K$'s minimum ropelength $L(K) \geq a\operatorname{Cr}(K)^{3/4}$

Definitions and mathematical statement

Let $K$ be a tame knot type represented by closed embedded curves in ${\mathbb{R}}^3$, and let $\operatorname{Cr}(K)$ be its crossing number, the minimum crossing count among generic planar diagrams. In the standard radius-based convention, the ropelength of a $C^{1,1}$ embedding $\gamma$ is \[ \operatorname{Rop}(\gamma)= \frac{\operatorname{length}(\gamma)}{\operatorname{thickness}(\gamma)}, \] where thickness is the maximal embedded normal-tube radius, equivalently the curve's reach in the usual convention. Define $L(K)$ as the infimum of this quantity over embeddings of type $K$.

The source asks for the constant in the universal inequality \[ L(K)\ge a\,\operatorname{Cr}(K)^{3/4}. \] Its research summary identifies the task as finding the optimal constant. A precise extremal expression is \[ a_* =\inf_{K:\operatorname{Cr}(K)>0} \frac{L(K)}{\operatorname{Cr}(K)^{3/4}}. \] The bound with coefficient $a_*$ is required for every knot type, not a coefficient chosen separately for each $K$. The unknot has crossing number zero and contributes no restriction to this ratio. The formula does not assume an extremizing knot realizes the infimum.

The supplied question reports $1.1<a\le10.76$ and the summary says existence of a positive coefficient and the exponent $3/4$ are established, while the sharp coefficient is unknown. The numerical range is retained as supplied evidence, not independently established here. Thickness as radius versus diameter changes the numerical coefficient; the source does not define its normalization, curve class, or minimum-versus-infimum convention. These must be matched before the displayed reported constants can be accepted in the chosen normalization.

Short English statement

Determine the best universal coefficient multiplying crossing number to the power $3/4$ in the minimum-ropelength lower bound. The stated interval $1.1<a\le10.76$ needs the source's precise thickness normalization.

The supplied summary attributes existence of a bound to Diao and improvements to Diao--Ernst and collaborators, but gives no exact paper or theorem. It reports no 2024--2026 determination of the sharp coefficient. This is a supplied partial-progress assessment, subject to the pinned Wikipedia record's primary-source review. The background's unpublished note conflicts with published registry metadata.

Sources

• Local UnsolvedMath registry, permanent ID 7200082, source code catalogue entry, ropelength/crossing-number constant question. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Wikipedia contributors, List of unsolved problems in mathematics, Geometry, bullet 82, pinned revision 1366636767, https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry; MediaWiki-source extraction, accessed 2026-07-29.
• Diao; Diao--Ernst and collaborators, as named in the research summary for the $3/4$ bound and improvements. No complete primary references or precise normalizations are supplied.
• Source-specific provenance: primary-source status audit NEEDS_REVIEW; background says unpublished pending audit although published metadata is true. Wikipedia text is CC BY-SA 4.0, with attribution and ShareAlike handling required for release, separately from dataset attribution.