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? The length of the curve divided by this thickness is called its ropelength. A knotted curve which achieves the minimum ropelength for its knot type is called a tight knot.
Applications of knot theory to the study of polymers like DNA have emphasized geometric measures for knotted curves, including this ropelength and various energies motivated by electrostatic potentials. Of course, another measure of complexity for a knotted curve is its average crossing number, the average number of crossings seen in the different planar projections of the curve. (This is of course always greater than the minimum crossing number for the knot type.)
The velocity of circular DNA strands (with a fixed number of basepairs) in an electrophoretic gel is correlated with the knot type of the strand, with more complicated knots moving faster. In fact, experiments reported in Nature in November 1996 suggested that there was a linear relation between the gel mobility and the average crossing number for a tight configuration of the knot. For the knot types considered, there even seemed to be a linear relation between this average crossing number and the ropelength.
Here, we show that such a linear relation between crossing number and ropelength cannot hold in general. Although there are families of knots (or links: for our purposes the number of components is unimportant) with increasing crossing number whose ropelength does grow linearly, other families can be built with less ropelength, growing in fact as the 3/4 power of crossing number. One such example is the family of Hopf links, and another is the family of (p,p-1)-torus knots.
Earlier arguments, which express the average crossing number as a double integral, and look at how much other rope can be near one strand, show that ropelength can never grow slower than this 3/4 power, so our examples are optimal.
It is unknown if any family of knots requires ropelength growing faster than linearly in crossing number, though it seems easy to rule out ropelength growing faster than quadratically.