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This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further condition that the ends of the rope can be smoothly connected.
In a mathematical setting, the rope who's ends are smoothly connected after knotting, can be modelled by a knotted channel surface with circular cross-sections of constant radius and, without self-penetrations.

What I would like to know, is:

  • how can the restrictions on knot-topology and being free of self-intersection be incorporated into an optimization problem for maximizing the "rope-thickness"?

  • do there already exist algorithms for solving those type of optimization problems?

  • would a solution to the problem yield knot-invariants like the maximal number of self-contacts?

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Jason Cantarella has studied this problem extensively, and written several papers on the topic. See his Octrope webpage:

Octrope is a library for quickly finding the thickness or ropelength of polygonal knots.

          Trefoil
          (Image from a Ridgerunner minimization animation for the trefoil, $3_1$.)

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  • $\begingroup$ Good pointer, Joseph! My problem seems to be known as the tight knot problem. $\endgroup$
    – Manfred Weis
    Commented Sep 9, 2015 at 12:32

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