I suspect what you are seeking is not known.
One of the experts in this area is Jason Cantarella, who coauthored (with R. Kusner and J. Sullivan) one of the definitive
papers on this topic, "On the minimum ropelength of knots and links," Inventiones Mathematicae,
Volume 150, Number 2, 257-286, 2002; Springer link).
Jason has developed software that he calls RidgeRunner which implements a rope-length minimization,
knot-tightening procedure. (I cannot speak to the details of his software, in particular,
I don't know if it is "akin to simulated annealing.") He maintains a wonderful web page
with (Quicktime) movies of his minimizer approaching what you call the "ideal knot" for many (more
than 100!) knots and links. Here, for example, is the endpoint of his optimizer for $8_5$:
Update. See the just-released paper, "The Shapes of Tight Composite Knots," arXiv:1110.3262 (math.DG), by Jason Cantarella and Al LaPointe and Eric Rawdon, for a description
of the RidgeRunner software mentioned above. It "proceeds by constrained gradient descent," and "is designed to stop at local minima of the ropelength function." They reduce the probability of "false local minima" with several
strategies. Their dataset now contains almost 1000 knots and links.
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I suspect what you are seeking is not known.
One of the experts in this area is Jason Cantarella, who coauthored (with R. Kusner and J. Sullivan) one of the definitive
papers in on this areatopic, "On the minimum ropelength of knots and links," Inventiones Mathematicae,
Volume 150, Number 2, 257-286, 2002; Springer link).
Jason has developed software that he calls RidgeRunner which implements a rope-length minimization,
knot-tightening procedure. (I cannot speak to the details of his software, in particular,
I don't know if it is "akin to simulated annealing.") He maintains a wonderful web page
with (Quicktime) movies of his minimizer approaching what you call the "ideal knot" for many (more
than 100!) knots and links. Here, for example, is the endpoint of his optimizer for $8_5$:
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I suspect what you are seeking is not known.
One of the experts in this area is Jason Cantarella, who coauthored (with R. Kusner and J. Sullivan) one of the definitive
papers in this area, "On the minimum ropelength of knots and links," Inventiones Mathematicae,
Volume 150, Number 2, 257-286, 2002; Springer link).
Jason has developed software that he calls RidgeRunner which implements a rope-length minimization,
knot-tightening procedure. (I cannot speak to the details of his software, in particular,
I don't know if it is "akin to simulated annealing.") He maintains a wonderful web page
with (Quicktime) movies of his minimizer approaching what you call the "ideal knot" for many (more
than 100!) knots and links. Here, for example, is the endpoint of his optimizer for $8_5$:
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