On Distortion and Thickness of Knots

by Robert B. Kusner & John M. Sullivan

Published in Topology and Geometry in Polymer Science, (S. G. Whittington, D. W. Sumners, T. Lodge, eds.), IMA Volumes in Math. and its Applications 103, Springer, 1998, pp 67-78. Plain-English Abstract

What length of rope (of given diameter) is required to tie a particular knot? Physical experiments suggest that about 16 inches are needed to tie a trefoil (overhand) knot in one-inch rope; other more complicated knots need more rope. Mathematically, we turn the problem around and ask, given a curve in space, how thick an embedded tube can be placed around the curve. That is, how thick a rope can be placed along the curve without self-intersection.

Intuitively, the diameter of the possible rope is bounded by the distance between strands at the closest crossing in the knot. But some work is needed to make this precise, since it is not clear when points along the knot should count as being on different "strands"; some points are close in space just because they are close along the knot.

Here, we give a new precise notion of thickness for space curves, based on Gromov's concept of distortion, and we relate this to another thickness defined in the literature. Our notion benefits from semicontinuity, so we should be able to prove the existence of thickest curves of prescribed length (or dually, shortest curves of prescribed thickness) in each knot class. These curves are of interest to chemists and biologists modeling polymers and DNA, because thickness may relate to the speed of knotted DNA strands in electrophoresis gels.

Note: In this paper, we gave a proof of Gromov's result that distortion is minimized by a round circle. We stated two conjectures about space curves, either of which would lead to a shorter proof. One conjecture was that any space curve has a planar unfolding, in which all chord-lengths have increased; we learned later that this had been proved in 1973 by G.T. Sallee.

The other conjecture was that any space curve of length L has a planar projection of diameter L/pi; this was proved in 2000 by Daniel Wienholtz in a series of preprints from Uni.Bonn. They are not available electronically, but my short summary of his arguments is online here.