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I have a way of generating random parameterized maps from $S^1 \to \mathbb{R}^3$. This method can create very simple knots, such as ellipses, but can also create knots with more crossings than I can count. The images can be knotted as I have seen the method generate a figure eight knot. I know that with a parameterization instead of a diagram one can compute things such as various knot energies. My question is the following: Is having the ability to generate random parameterized knots of use to anyone?

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Not a comment on the question itself but on the peculiar invented word you used: "parameterize". Statisticians talk about "parametric statistics", not "parameteric". But dictionaries don't help with the spelling. The usual pronunciation of "parametrization" is similar to "metrization". No one writes "meterization". (My credentials? I'm the founder but so far only member of the Foundation to Stamp Out Indeterminants.) – Jim Humphreys Jul 16 '10 at 21:58
A computationally efficient method that finds lots of interesting knots could conceivably be folded into something like SnapPea, as a way to generate knots to test invariants against and measure run times, etc. In any case, this post reads as much like an advertisement as many I've read on MO. – Theo Johnson-Freyd Jul 16 '10 at 22:20
Jim, I pronounce it "parameterization," with a stress on the am, though I swallow the syllable in "paramet(e)rize." I think that the difference is that I see metrization as moving to the metric system, not moving to the meter. Similarly, symmetrization comes from symmetric and geometrization comes from geometric or geometry. Perhaps we can compromise with "parametry" in place of -ization? (ps, no relation) – Ben Wieland Jul 16 '10 at 22:32
up vote 1 down vote accepted

Not very likely. Any physical questions about random knotting will come with all sorts of requirements about the random process that generates the knots, and so it's unlikely that whatever method you have in mind would be helpful in any given situation.

Also, it's very easy to produce random knots, and without specifying some nice property your method has it's hard to say much. The simplest method I can think of is to pick n random directions in R^3, and concatenate a series of n unit intervals in those directions, then join the first and last point. Why should your method being any more or less interesting than this one?

You might be interested in reading Dorothy Buck's papers on the arXiv, which include some simple models for transformations of knots that are relevant for knotted DNA, to get an idea of the very particular requirements of specific applications of "random knotting".

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I guess the only nice property is that the maps are smooth, but your method can be smoothed as well. This was kind of the answer I was expecting after looking around a bit. Thanks! – user7681 Jul 16 '10 at 22:27

You may be interested in the answers to this MathOverflow question.

In particular in the paper by Lubotzky, Maher and Wu cited there, there is an application of random 3-manifold theory to a non-random question. Thus in principle a model for generating random knots might have an application to knot theory itself, such as proving that there exist infinitely many knots with such and such properties.

As for applications to physics or biology, I agree with Scott Morrison's answer.

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