Abstract
Applications of knots to the study of polymers have emphasized geometric measures on curves such as ‘energy’1,2,3,4 and ‘rope length’5,6,7, which, when minimized over different configurations of a knot, give computable knot invariants related to physical quantities8. In DNA knots, electrophoretic mobility appears to be correlated with the average crossing number of rope-length-minimizing configurations9, and a roughly linear empirical relation has been observed between the crossing number and rope length10. Here we show that a linear relation cannot hold in general, and we construct infinite families of knots whose rope length grows as the 3/4 power of the crossing number11. It can be shown that no smaller power is possible12,13,14.
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Cantarella, J., Kusner, R. & Sullivan, J. Tight knot values deviate from linear relations. Nature 392, 237–238 (1998). https://doi.org/10.1038/32558
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DOI: https://doi.org/10.1038/32558
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