What is the maximum shortening of a string after tying an overhand knot?

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SUMMARY

The maximum shortening of a string after tying an overhand knot is determined by its physical properties, specifically its radius R. The absolute minimum length/thickness for any type of knot is established at 15.66, while the trefoil knot specifically has a maximum shortening of 16.372. These values are derived from the concept of ropelength, which is influenced by the knot's crossing number, particularly noted as 3 for the trefoil knot. Further exploration of knot projections may yield additional insights into this phenomenon.

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Given a flexible, inelastic and incompressible string of radius R, how much at most would it shorten from tying an overhand knot in it?
 
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The following is not the answer to the question you asked, but try this for starters:

The http://en.wikipedia.org/wiki/Overhand_knot" .

http://en.wikipedia.org/wiki/Ropelength" is expressed as a ratio to thickness and depends heavily (for simple knots) on crossing number (3 for trefoil).

The absolute minimum length/thickness for any type of knot was shown to be 15.66, and for the trefoil specifically the answer is 16.372.
 
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These are great lines, sneutron! Where did you get them?

Could projections of a knot's perpendicular surfaces (while the linear string rotates) give any insight?
 
Wiki, Loren, just follow the links in my post.
 

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