Knot security (When to trust your life with a knot)
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:
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).
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..