The Distortion of a Knotted Curve
Geometric Topology
2007-12-29 v3 Differential Geometry
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.
Cite
@article{arxiv.math/0409438,
title = {The Distortion of a Knotted Curve},
author = {Elizabeth Denne and John M Sullivan},
journal= {arXiv preprint arXiv:math/0409438},
year = {2007}
}
Comments
5 pages, 7 figures; substantial improvements: bound is now 5pi/3 and applies to all tame knots