The distortion of a knotted curve

Denne, Elizabeth; Sullivan, John

doi:10.1090/S0002-9939-08-09705-0

The distortion of a knotted curve
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by Elizabeth Denne and John M. Sullivan PDF

Proc. Amer. Math. Soc. 137 (2009), 1139-1148 Request permission

Abstract:

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed that 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 $5\pi /3$; 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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