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The Distortion of a Knotted Curve

Elizabeth Denne·John M Sullivan
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Abstract

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil knot. Our argument uses the existence of a shortest essential secant and a characterization of borderline-essential arcs.

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Elizabeth Denne, John M Sullivan. 2007-12-29. The Distortion of a Knotted Curve. https://arxiv.org/abs/math/0409438

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