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

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Discussion Overview

The discussion revolves around the maximum shortening of a flexible, inelastic, and incompressible string when tied into an overhand knot. Participants explore theoretical aspects and implications of knot geometry, particularly focusing on the relationship between knot structure and string length.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks about the maximum shortening of a string when tied into an overhand knot, specifying the properties of the string.
  • Another participant references knot theory, mentioning the concept of ropelength and its dependence on knot thickness and crossing number, providing specific numerical values for the trefoil knot.
  • A third participant inquires about the potential insights from projecting a knot's perpendicular surfaces as the string rotates, suggesting a geometric approach to understanding the problem.
  • A later reply directs the inquiry back to the Wikipedia sources for further information on knots and ropelength.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the maximum shortening of the string, and multiple perspectives on knot theory and geometry are presented without resolution.

Contextual Notes

The discussion includes references to specific knot properties and mathematical concepts, but lacks detailed exploration of the assumptions behind the calculations or the implications of the mentioned values.

Who May Find This Useful

Individuals interested in knot theory, geometry of strings, and mathematical modeling of physical properties of knots may find this discussion relevant.

Loren Booda
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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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