The distortion of a knotted curve

2008/09/29 by Elizabeth Denne, John M. Sullivan · 2 citations
Mathematics · Engineering · Computer Science · #Geometric and Algebraic Topology #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation

paper · pdf · doi:10.1090/s0002-9939-08-09705-0

Abstract

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed that any closed curve has distortion at least π /2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5π /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.

Cited by

• Extrinsic systole of Seifert surfaces and distortion of knots
• On the question of genericity of hyperbolic knots

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