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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The distortion of a knotted curve


Authors: Elizabeth Denne and John M. Sullivan
Journal: Proc. Amer. Math. Soc. 137 (2009), 1139-1148
MSC (2000): Primary 57M25; Secondary 49Q10, 53A04, 53C42
Published electronically: September 29, 2008
MathSciNet review: 2457456
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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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Additional Information

Elizabeth Denne
Affiliation: Department of Mathematics and Statistics, Smith College, Northampton, Massachusetts 01063
Email: edenne@email.smith.edu

John M. Sullivan
Affiliation: Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
Email: jms@isama.org

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09705-0
PII: S 0002-9939(08)09705-0
Received by editor(s): February 28, 2008
Published electronically: September 29, 2008
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.