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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 ( http://en.wikipedia.org/wiki/Tangle_(mathematics) ).2012-10-20
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    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.2012-10-20
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    You can also see this site, the references there and the knot atlas. http://en.wikipedia.org/wiki/Knot_(mathematics)2012-10-20
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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.2012-10-20
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    @MJD: I didn't claim there was. But your "from the point of view of knot theory" sentence is misleading. There is an interesting topological classification of tangles and it doesn't imply that all of the examples you've given are equivalent. Anyway, this doesn't seem like primarily a mathematical question to me: it's some combination of physics, mechanics, maybe material science...2012-10-20
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    @QiaochuYuan Thanks. I have deleted that paragraph.2012-10-20
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    I was surprised to find that there is no [tag:mechanics] tag. (There is [tag:classical-mechanics], which is not applicable here.) Rather than create one, I retagged the question with the less specific [tag:physics].2012-10-20
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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.2012-10-20
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    I forgot about "[Spontaneous knotting of an agitated string](http://www.pnas.org/content/104/42/16432.full)", which is somewhat relevant.2012-10-20
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    Also quasi-relevant: [Tie knots, random walks and topology](http://www.tcm.phy.cam.ac.uk/~tmf20/TIES/PAPERS/paper_physica_a.pdf), by Fink and Yao.2012-10-21
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    Possibly relevant: [Mechanical engineering study at UCB](http://news.berkeley.edu/2017/04/11/shoe-string-theory-science-shows-why-shoelaces-come-untied/)2017-04-12
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    Relevant: [The roles of impact and inertia in the failure of a shoelace knot](http://rspa.royalsocietypublishing.org/content/473/2200/20160770)2017-06-09

3 Answers 3

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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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Untangling the mechanics and topology in the frictional response of long overhand elastic knots by Jawed et al, accepted for publication in Physical Review Letters, seems to be relevant to this topic. I will add a link top the paper if one becomes available.

Forget Dark Energy: MIT Physicists Have Finally Cracked Overhand Knots has a pop discussion of the research.