The Distortion of a Knotted Curve

TL;DR

This paper establishes a lower bound of 5pi/3 for the distortion of any nontrivial tame knot, advancing understanding of geometric properties of knots and providing bounds relevant for knot construction.

Contribution

It proves a new lower bound for knot distortion and characterizes essential arcs, improving theoretical understanding of knot geometry.

Findings

Nontrivial tame knots have distortion at least 5pi/3

Distortion under 7.16 can be used to construct a trefoil knot

Existence of shortest essential secant is key to the proof

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.