From John M. Sullivan, University of Illinois
Urbana
Keith Devlin describes knots that minimise the length of rope they
require—the quotient of length by thickness
(10 November, p 40).
He says that “no one has ever proved” that each type of knot has such a tight (or
“ideal”) configuration. My paper with Jason Cantarella and Rob Kusner, “On the
Minimum Ropelength of Knots and Links” (www.math.uiuc.edu/~jms/ Papers/)
proves that they do.
For each knot or link type, we show rigorously that there is a ropelength
minimiser, which must have a certain (low) degree of smoothness. We also give
the first families of explicit examples of tight links. These examples
demonstrate that no greater smoothness can be expected in general for tight
links. In some cases, however, we describe not a unique tight configuration, but
instead a whole continuous family of equal minimisers of different shapes.
