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.)