You may not have realized, but when tying shoelaces, you're employing a handy combination of topology and elasticity. Despite being such a routine thing, understanding how shoelace — and other — knots work is far from trivial, because of the many physical parameters involved.
Khalid Jawed and colleagues have now started tackling the physics of simple overhand knots. Such knots consist of a loop, a braid with a certain uneven number, 2n + 1, of crossings and two ends (pictured above for n = 1). The authors made macroscopic knots from Nitinol, a highly elastic nickel titanium alloy, and measured the force needed to pull the knotted rope over a given distance. They also imaged how the knot's shape changes when pulled.
Jawed et al. were able to explain the resulting data in terms of a model that properly takes into account the bending stiffness of the ropes and draws on Gustav Kirchhoff's treatment of the elastic rod. They found that the applied tensile force depended on the end-to-end shortening of the rope in a nonlinear way, and that the dependence changed drastically with the unknotting number n.
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Verberck, B. Knot so simple. Nature Phys 11, 800 (2015). https://doi.org/10.1038/nphys3514
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DOI: https://doi.org/10.1038/nphys3514
