On Distortion and Thickness of Knots

What length of rope (of given diameter) is required to tie a particular knot? Or, to turn the problem around, given an embedded curve, how thick a regular neighborhood of the curve also is embedded? Intuitively, the diameter of the possible rope is bounded by the distance between strands at the closest crossing in the knot. But of course the distance between two points along a curve goes to zero as the points approach each other, so to make the notion precise, we need to exclude some neighborhood of the diagonal. Various notions of thickness have been proposed recently. For example, [LSDR] defines a thickness by considering the distance function between points on the curve only where it has critical points. But the definition there also involves the minimum radius of curvature of the curve, and thus is unbounded for polygonal curves. Here we introduce and compare two new families of thickness measures. One makes use of Gromov’s concept of distortion (see [Gro1] and [Gro2, p. 114] and [GLP, pp. 6–9]); it applies to all rectifiable curves (including polygons). The other generalizes the notion from [LSDR]. Our main result is a basic inequality (Theorem 5.2) between these measures. The distortion thickness should permit us to prove the existence of thickest curves of prescribed length (or dually, shortest curves of prescribed thickness) in each knot class; such curves are of interest to chemists and biologists modeling polymers and DNA (see, for example, [KBM]). Moreover, curvature bounds should follow from the optimality, and we conjecture that, for an optimal knot, all reasonable measures of thickness should be equal.