Take a piece of ideal rope: of constant radius, ideally flexible, and completely slippery. Tie a tight knot into it, as shown in the figure below.
By how much did the two ends of the rope come closer together?
2026-03-29 20:27:41.1774816061
Ropelength of tight knot
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Suppose that the rope has radius $1$. The shortest length of a radius $1$ rope that can make a given knot is called the rope length of the knot. In this formulation, the knot is a closed rope, not one with open ends as in your picture. If one closes the ends of your rope, then the knot obtained is the trefoil knot.
The shortest rope length for a trefoil that I could find in the literature is 32.7433864 (see here). The best lower bound for the trefoil that I could find in the literature is 21.45 (see here). So there is quite a bit of difference between the tightest trefoil we can make and the provable lower bound for the tightest trefoil possible. Given the status of the problem for closed knots, it is unlikely that an exact answer is known for your question for the trefoil with ends.
Perhaps one could use the bounds in the previous paragraph on rope length of the closed trefoil to derive bounds on the quantity requested in your formulation of the problem.