The Distortion of a Knotted Curve

Details The Distortion of a Knotted Curve The Distortion of a Knotted Curve

2004-09-22

arxiv.org/abs/math/0409438v3

Mathematics

Geometric Topology

5 pages, 7 figures; substantial improvements: bound is now 5pi/3 and applies to all tame knots

Scientific paper

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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