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This question is related to a a question about self-tightening knots.

I am supervising a senior thesis and my student is interested in knots. My student is also a rock climber and has an interesting idea for a project, but I am not sure how reasonable it is. She wants to look at the mathematics of knot security. As a climber, she may tie a single rope to a closed loop or tie two pieces of rope together. There are some knots you should trust your life with and others you should avoid while hanging from a cliff. She wants to understand what properties of a knot make it secure. I like the idea, but I have no idea what kind of model we would use. The only articles I could find were about knot security for surgeons tying sutures and were based on experimental evidence only.

Does anyone know any mathematical references for studying physical knots and knot security?

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I recommend some "senior thesis security". If this does not pan out, what could she do instead that is related? For example, is there work that talks about the energy involved in forming or untying a knot? Gerhard "Always Mount A Scratch Thesis" Paseman, 2012.09.17 –  Gerhard Paseman Sep 17 '12 at 22:32
I could see this depending heavily on the diameter of the rope, because changing this would change the points of contact and so change the pressure on each point. It could also be that two (topologically) identical knots with identical rope have different points of contact resulting in very different strengths. –  Alex Becker Sep 17 '12 at 23:01
Maybe this paper can help: Alexander Coward, Joel Hass, Topological and physical knot theory are distinct (arxiv.org/abs/1203.4019) –  Adrien Sep 17 '12 at 23:04
This may be helpful: allaboutknots.blogspot.com/2010/11/… The fundamental physical fact to understand about knots is that as a knot wraps around a cylinder (such as a tree branch, or another piece of rope), the maximum tension supportable by friction goes up as $\exp(\mu \theta)$, where $\mu$ is the coefficient of static or kinetic friction. So there are at least some cases where the analysis is independent of the detailed structure of the rope (cf. @Alex Becker). If it's not always scale-invariant, perhaps there are scaling laws? –  Ben Crowell Sep 17 '12 at 23:39
A relevant question: mathoverflow.net/questions/85186/self-tightening-knot –  Anton Petrunin Sep 18 '12 at 11:48

2 Answers 2

up vote 16 down vote accepted

This is not a definitive answer to your query, but you might be interested in the model explored in the paper,

  • "Localization of Breakage Points in Knotted Strings," Piotr Pieranski, Sandor Kasas, Giovanni Dietler, Jacques Dubochet, and Andrzej Stasiak, New Journal of Physics, vol. 3, June 2001, p. 1-13 (journal link).

They model a knot by a curve with points surrounded by hard spheres, akin to a polymer chain, and then obtain results somewhat in agreement with physical experiments. This image gives you a sense of their work:
     Trefoil in tension

In the case of knotted spaghetti, the breakage occurs at the position with high curvature at the entry to the knot. [...] Rock climbers and anglers know that a simple overhand knot tied on a mountaineering rope or a fishing line weakens it substantially.

Two references that address friction (Ben Crowell's comment):

  • Maddocks, J.H. and Keller, J. B., "Ropes in Equilibrium," SIAM J Appl. Math., 47 (1987), pp. 1185-1200. (Journal link).
    Abstract. The mechanical equilibrium of a rope wrapped around a solid body or around another rope is investigated, with friction and tension being taken into account. Various examples are treated to illustrate the theory. Its application to knots and hitches is indicated.

  • "An Introduction to the Theory of Hitches and Knots," Jeremy Stolarz, (PDF link).
              Clove Hitch

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Very cool! But there are two qualitatively different ways that a knot can fail. (1) The rope breaks. (2) The knot slips. This figure shows an example in which friction is irrelevant, and the only mode of failure is #1. For the application stated by the OP, the relevant mode of failure is #2. A fairly physically transparent example is the Prusik knot: en.wikipedia.org/wiki/Prusik (Whether #1 or #2 is more fun mathematically is a whole different matter...) –  Ben Crowell Sep 18 '12 at 3:42
Thank you very much for the references! –  b b Sep 18 '12 at 21:12
The Stolarz paper seems to be exactly what the OP had in mind. There seem to be some errors, though. E.g., the condition $\mu \ge (1/2)e^{\pi\mu}$ given after eq. 22 has no real solutions, which would imply that a square knot can never hold. –  Ben Crowell Sep 19 '12 at 5:33
Two more references: Maddocks, J.H. and Keller, J. B., "Ropes in Equilibrium," SIAM J Appl. Math., 47 (1987), pp. 1185-1200. Jearl Walker (Amateur Scientist column), "In which simple equations show whether a knot will hold or slip," Sci Am 249:2, p. 120, August 1983. Maddocks gives the condition for a square knot to hold as $1 \le 2\mu e^{\pi\mu}$. This makes more sense than Stolarz's relation, since it can hold or fail for different values of $\mu$. The critical value of $\mu$ is about 0.237, which is physically reasonable. –  Ben Crowell Nov 8 '12 at 15:50

Louis Kauffman's masterful book "Knots and Physics" has some thoughts on it. Especially the introductory chapter, and then later a small chapter called "The Theory of Hitches". Worth a reading IMO..

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Kauffman's discussion of physical knots in ropes is extremely brief. The student paper by Stolarz (linked to from Joseph O'Rourke's answer) has an exposition more detailed than Kauffman's. They all seem to be referring to Bayman, Am J Phys, 45 (1977) 185, which I haven't seen yet. –  Ben Crowell Nov 7 '12 at 16:26

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