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The ropelength of a knot curve $C$ is defined as the ratio $L(C) = \lambda(C)/ \tau(C)$, where $\lambda(C)$ is the length of $C$ and $\tau(C)$ is the thickness of the knot defined by $C$ [from Wikipedia]. Intuitively the thickness is the maximally possible thickness that $C$ as a rope can have (without intersecting itself).

The infimum ropelength of the realizations of a knot type is a knot invariant. A knot that minimizes ropelength is called an ideal knot.

Question 1: How can it be proved that there are ideal knots at all (besides the plane circle which is the ideal unknot)?

Another question concerns the number (of shapes) of ideal knots of a given knot type.

Question 2: Is the number of ideal knots known, at least for some knot typyes? Can it be calculated from other knot invariants, eventually?

(Assuming that this number is finite, it would be another knot invariant – the more interesting the more it can vary.)

Question 3: Are there ideal knots of constant curvature (besides the plane circle)?

Alternatively: Are there approximations of ideal knots that approximate constant curvature?

(The last one is a refinement of another question.)

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1 Answer 1

Questions 1 and 2 are addressed in the paper "On the Minimum Ropelength of Knots and Links" by Cantarella, Kusner and Sullivan, which is cited in the wikipedia article on ropelength.

The proof of the existence (and $C^{1,1}$ regularity) of minimizers is in Section 2 and is an application of the direct method.

Some examples pertinent to your question about the uniqueness of minimizers are discussed at the end of section 3. Apparently there is a 1-parameter family of minimizers of a five-component link with length $2\pi+8$. I am not sure if there are any examples of non-uniqueness for knots though.

Jason Cantarella has probably written the most on finding numerical approximations to ropelength critical and ropelength minimal knots and links so he may be able to answer question 3. See for instance this paper on the shapes of ideal composite knots.

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See also the earlier MO question, "Is it possible to reliably generate a particular approximation of an ideal knot via a simulated annealing approach?," which also cites the software behind the numerical approximations jc mentions: mathoverflow.net/questions/76545/… –  Joseph O'Rourke Feb 28 '13 at 2:35

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