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Topological mechanics of knots and tangles

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Science  03 Jan 2020:
Vol. 367, Issue 6473, pp. 71-75
DOI: 10.1126/science.aaz0135

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It's knot what you know

Why is it that some knots seem to hold tight while others readily slip apart? Patil et al. develop a theoretical analysis of the stability of knots and find links between topological parameters (twist charge, crossing numbers, handedness) and mechanical stability. The theory is confirmed using simulations and experiments on color-changing fibers that optically show localized stress differences in different parts of the knot as the two strands are pulled apart. The authors show why some common knots slip easily and untie, whereas others hold tight.

Science, this issue p. 71

Abstract

Knots play a fundamental role in the dynamics of biological and physical systems, from DNA to turbulent plasmas, as well as in climbing, weaving, sailing, and surgery. Despite having been studied for centuries, the subtle interplay between topology and mechanics in elastic knots remains poorly understood. Here, we combined optomechanical experiments with theory and simulations to analyze knotted fibers that change their color under mechanical deformations. Exploiting an analogy with long-range ferromagnetic spin systems, we identified simple topological counting rules to predict the relative mechanical stability of knots and tangles, in agreement with simulations and experiments for commonly used climbing and sailing bends. Our results highlight the importance of twist and writhe in unknotting processes, providing guidance for the control of systems with complex entanglements.

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