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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#3
Loren Booda
3,108
4
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?