The Distortion of a Knotted Curve

2004/09/22 by Elizabeth Denne, Denne, Elizabeth, John M Sullivan +1 · 1 citation
Mathematics · #49Q10 #53A04 #53C42 (Secondary) #57M25 (Primary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.DG #math.GT #msc:49Q10 #msc:53A04 #msc:53C42 #msc:57M25

paper · pdf · doi:10.48550/arxiv.math/0409438

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

arxiv created 2007/12/29 · arxiv updated 2009/12/01

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.

Cited by

• Extrinsic systole of Seifert surfaces and distortion of knots

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• The distortion of a knotted curve
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• New lower bound for the distortion of a knotted curve
• Extrinsic systole of Seifert surfaces and distortion of knots
• Quadrisecants and essential secants of knots: with applications to the geometry of knots