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What space curve defines a Trefoil knot made of ideal flexible nonstretchable steel wire? By ideal flexible nonstretchable steel wire I mean wire that meets this requirements:

  1. a wire cannot shorten or extend but can be bend without limit

  2. a wire tends to straighten itself in every point

  3. a wire has zero friction with itself.

Curve can be defined explicitly, implicitly, parametrized or numerically.

EDIT: I uploaded an example of torus knot made of steel wire to show that the shape is unique. You can squeeze it and it returns into the same shape (of course the wire is real world and not ideal, but you can get the idea).

example of torus knot

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  • $\begingroup$ It can be found on the surface of a torus, which gives one reasonably nice representation: en.m.wikipedia.org/wiki/Torus_knot $\endgroup$
    – pjs36
    Commented Mar 26, 2017 at 21:19
  • $\begingroup$ @pjs36 How do we know it is the case? There are many different parametric curves that produce Trefoil knot. Why you think the one on torus satisfy my requirements? $\endgroup$ Commented Mar 26, 2017 at 21:25
  • $\begingroup$ I misunderstood what you were looking for, I don't have any reason to believe that one works. $\endgroup$
    – pjs36
    Commented Mar 26, 2017 at 21:38
  • $\begingroup$ (+1), but it looks to me that one needs to specify explicitly how "energy" depends on curvature and torsion and, as in your comment below, attempt to minimize the integral of "energy" while fixing the isotopy class of the image. (Incidentally, it's not obvious to me that the physical object pictured is an absolute minimum unless the object is modeled as a tube of finite cross section whose energy depends on the twisting; a curve in this general spatial shape might be possible to make "nearly planar", the way a trefoil might converge to a loop traced three times.) $\endgroup$ Commented Mar 28, 2017 at 19:16
  • $\begingroup$ @ Andrew D. Hwang: Yes, if the diameter of the wire is infinitesimal then there is another shape, that satisfy my requirements and as you said it is planar loop traced three times. But I would call it a trivial solution. There may exist several other nontrivial solutions but I believe there is only one another solution, but of course I can not prove it. I also thought torsion would be zero through the wire and "energy" only depends on curvature. I am not sure whether it is possible to have zero torsion on the whole curve. $\endgroup$ Commented Mar 28, 2017 at 20:23

2 Answers 2

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Another place to look:

Kauffman, Louis H. "Following Knots Down Their Energy Gradients." Symmetry 4.2 (2012): 276-284. (arXiv abs.)


          Fig1


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    $\begingroup$ Looks like I am going to experiment with KnotPlot. $\endgroup$ Commented Mar 28, 2017 at 21:57
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Perhaps ropelength minimizers are close to your requirements?


IdealTref
Image from Wikipedia Ropelength article.


See, e.g.,

Cantarella, Jason, Robert B. Kusner, and John M. Sullivan. "On the minimum ropelength of knots and links." Inventiones mathematicae 150.2 (2002): 257-286. Journal link.

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  • $\begingroup$ The length is irrelevant for me. Say you are given 1 meter of ideal steel wire of 1mm width (or could be infinitesimally small). What would be the space curve Trefoil knot made of this wire? $\endgroup$ Commented Mar 26, 2017 at 22:22
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    $\begingroup$ @azerbajdzan: I don't believe your conditions lead to a unique answer. There are an infinite number of trefoils satisfying those conditions, as far as I understand them. You need something else, such as: minimum length, minimum energy, ... ? $\endgroup$ Commented Mar 26, 2017 at 22:28
  • $\begingroup$ I marked your answer as useful, because I think the middle of the rope in your image is the curve I am looking for. But they say it is a numerical approximation but there is no explanation how they computed it. $\endgroup$ Commented Mar 26, 2017 at 22:28
  • $\begingroup$ @azerbajdzan: There is an entire theory behind that image, several technical papers. I'll add one to my post. $\endgroup$ Commented Mar 26, 2017 at 22:30
  • $\begingroup$ I think it is unique. For example if we had a simple loop of ideal wire it would form into circle, the only possible shape that meets my requirements. $\endgroup$ Commented Mar 26, 2017 at 22:32

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