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The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled outwards, requires some extra rope length compared to three ropes that are not intertwined/linked/tangled.

symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1

The extra rope length is zero for an infinite size of the inner triangle. The extra rope length increases when the inner triangle gets smaller.

How exactly does the extra rope length depend on the size of the inner triangle?

Clarification: the rope is assumed to always have a circular cross section, even if bent, and the rope length is defined as the length of the centerline of all circles. (This is common in knot theory.) ((The size of the inner triangle is not a uniquely defined quantity, especially if the ropes are not straight. Any definition is ok.))

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  • $\begingroup$ I don't get it - isn't this simply $3(\sqrt{L^2 +4} -L)$ if we denote the triangle side length by $L$? $\endgroup$
    – Michael Engelhardt
    Commented Oct 29, 2023 at 1:43
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    $\begingroup$ I think you will have to define the very notion of the "extra rope length", because the ropes will be of some non-cylindrical form, which is probably very complicated and depending on mechanical properties of the material of the rope. $\endgroup$
    – Iosif Pinelis
    Commented Oct 29, 2023 at 1:55
  • $\begingroup$ Yes, the form is non-cylindrical. No, it does not depend on the material if the rope is always of circular cross section. Yes the from is complicated, because the radius is not neglegible compared to L. $\endgroup$
    – Claudio
    Commented Oct 29, 2023 at 3:54
  • $\begingroup$ The ropes are bent, and that is the hard part of the problem. The formula by Michael is an approximation that neglects the bends. It is not even clear whether, seen from above, the solution looks like three straight lines. $\endgroup$
    – Claudio
    Commented Oct 29, 2023 at 4:02
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    $\begingroup$ I get it even less than before. If you want to get into that much detail (more detail than the approximation I gave), then you have to specify the problem more. Apparently, the rope material is highly anisotropic (since it can go around a corner without losing circular cross section). Is it even describable by an elasticity tensor, or is it nonlinear? How strongly are we pulling on it? I agree with Iosif that, at this level of detail, more information on the mechanical properties is needed. You speak of solution. Solution to what? What is the dynamical equation governing the rope? $\endgroup$
    – Michael Engelhardt
    Commented Oct 29, 2023 at 6:50

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