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From the point of view of people tying real knots (canonically, sailors) mathematical knot theory ignores much of what makes the problem of knot-tying interesting. Some matters that come up in the study of physical knots include:

  1. To tie two ropes together, a sheet bend is strongly preferable to a square knot, because the latter tends to capsize.
  2. The grass bend is extremely prone to slip when tied in cord or rope, but is much safer when tied in flat belts.
  3. The clove hitch is good for round posts, but unreliable when tied around a square post.
  4. The constrictor knot, although similar to the clove hitch, is much more difficult to untie.
  5. The bowline is easy to untie when wet, but the water bowline is even easier to untie when wet.

Is there any sort of mathematical analysis of knots that predicts properties like these?

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You don't give knot theorists enough credit! Your examples are known in mathematical lingo as tangles ( en.wikipedia.org/wiki/Tangle_(mathematics) ). –  Qiaochu Yuan Oct 20 '12 at 22:27
    
The reason why some of this knots are not interesting is because according to the definitions in knot theory they are not knots at all. –  glebovg Oct 20 '12 at 22:27
    
You can also see this site, the references there and the knot atlas. en.wikipedia.org/wiki/Knot_(mathematics) –  Amzoti Oct 20 '12 at 22:28
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@Qiaochu I'm aware of tangles, but if there is any application of tangles to the sort of questions I listed, I have not seen it. –  MJD Oct 20 '12 at 22:29
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Physical knot theory might be what you are looking for. It is essentially a study of knots with physical properties, hence physical knot theory describes more realistic knots. –  glebovg Oct 20 '12 at 22:46

2 Answers 2

I don't know the explicit content of this course at MIT, but it may be of interest:

http://math.mit.edu/~ormsby/knots.html

And here is a link to a piece about Fields Medal winner Vaughan Jones describing his work on knots. It's a bit removed from your practical questions, but clearly reflects a link between physics and knots

http://blogs.vanderbilt.edu/research/2011/04/beyond-knot-theory/

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http://en.wikipedia.org/wiki/Physical_knot_theory

http://en.wikipedia.org/wiki/Knot_energy

I suspect that the theory is not well-developed as it applies to the examples in the posted question.

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