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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

azerbajdzan
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  • It can be found on the surface of a torus, which gives one reasonably nice representation: https://en.m.wikipedia.org/wiki/Torus_knot – pjs36 Mar 26 '17 at 21:19
  • @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? – azerbajdzan Mar 26 '17 at 21:25
  • I misunderstood what you were looking for, I don't have any reason to believe that one works. – pjs36 Mar 26 '17 at 21:38
  • (+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.) – Andrew D. Hwang Mar 28 '17 at 19:16
  • @ 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. – azerbajdzan Mar 28 '17 at 20:23
  • By zero torsion I mean that the wire is not twisted about its longitudinal axis. – azerbajdzan Mar 28 '17 at 20:30

2 Answers2

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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
Joseph O'Rourke
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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.

Joseph O'Rourke
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  • 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? – azerbajdzan Mar 26 '17 at 22:22
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    @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, ... ? – Joseph O'Rourke Mar 26 '17 at 22:28
  • 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. – azerbajdzan Mar 26 '17 at 22:28
  • @azerbajdzan: There is an entire theory behind that image, several technical papers. I'll add one to my post. – Joseph O'Rourke Mar 26 '17 at 22:30
  • 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. – azerbajdzan Mar 26 '17 at 22:32
  • ...it would not form into, say, ellipse, because my condition 2 says: "a wire tends to straighten itself in every point". So in points with bigger curvature the loop would straighten until it forms circle. – azerbajdzan Mar 26 '17 at 22:38
  • @azerbajdzan: So you are minimizing the maximum curvature. – Joseph O'Rourke Mar 26 '17 at 22:40
  • Perhaps yes, or maybe minimizing total absolute curvature is more precise. – azerbajdzan Mar 26 '17 at 22:49
  • I thought about the image of ideal length knot from wikipedia again. I think it would form from an ideal rubber rope. By ideal rubber rope I mean a rope that tends to shorten itself in every point until it can not anymore and its diameter is always constant. So now I think minimal length knot is not the same as mine. – azerbajdzan Mar 26 '17 at 22:59