On Distortion and Thickness of Knots
Robert B. Kusner and John M. Sullivan
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.
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.
Robert B. Kusner email@example.com http://www.gang.umass.edu/~kusner/ John M. Sullivan firstname.lastname@example.org http://www.math.uiuc.edu/~jms/