Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it

Definitions and mathematical statement

The source asks how much the endpoint separation of a long radius-one rope decreases when a tight open knot is tied. To distinguish the quantities, for an inextensible rope with centerline $\gamma:[0,L]\to\mathbb R^3$ parametrized by arclength, define \[ d(\gamma)=|\gamma(L)-\gamma(0)|,\qquad \Delta(\gamma)=L-d(\gamma). \] The reference straight rope has separation $L$. Radius $1$ would ordinarily be modeled by a non-self-overlapping unit-thickness tube, with a precise touching/end-cap convention; the source does not supply that model.

A possible tension-tightening formulation, not a source-specified definition, would choose an admissible class $\mathcal C_K(L)$ of unit-thickness curves of the intended open-knot type $K$ and ask for \[ \Delta_K(L)=L-\sup_{\gamma\in\mathcal C_K(L)}d(\gamma), \] or for a sufficiently-long-rope limit of this quantity. The excerpt does not select this optimization, a mechanical equilibrium, or another meaning of "tight", and does not state whether a formula may depend on $K$, $L$, or additional parameters.

An open arc with freely moving ends does not carry the same fixed knot type as a closed loop. Endpoint constraints, a chosen closure, or a long-knot convention such as straight behavior outside a compact set are needed before $\mathcal C_K(L)$ is meaningful. The source gives none, and also does not specify stretching, friction, tail directions, or what "long" requires. No closed-knot ropelength minimization is silently substituted for this endpoint-shortening question.

The summary reports no verified general expression and points to related tight-knot literature, not a precise theorem answering this question. The pinned Wikipedia open label and search report remain source assessments pending primary-source review.

Short English statement

Find the endpoint shortening caused by a tight open knot in a radius-one rope. A precise answer needs the knot convention, rope length and thickness model, endpoint constraints, and definition of tightening; the source supplies none of these choices.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $7200084$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Wikipedia contributors, List of unsolved problems in mathematics, Geometry bullet 84, 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; primary-source status review remains required.
• "Knot Tightening by Constrained Gradient Descent", arXiv:1002.1723; "The Shapes of Tight Composite Knots", arXiv:1110.3262. These are supplied as related literature, without authors or an asserted general shortening formula.