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.References
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Additional Information
- Elizabeth Denne
- Affiliation: Department of Mathematics and Statistics, Smith College, Northampton, Massachusetts 01063
- MR Author ID: 780494
- Email: edenne@email.smith.edu
- John M. Sullivan
- Affiliation: Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
- Email: jms@isama.org
- Received by editor(s): February 28, 2008
- Published electronically: September 29, 2008
- Communicated by: Daniel Ruberman
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1139-1148
- MSC (2000): Primary 57M25; Secondary 49Q10, 53A04, 53C42
- DOI: https://doi.org/10.1090/S0002-9939-08-09705-0
- MathSciNet review: 2457456