Möbius invariance of knot energy

Abstract

A physically natural potential energy for simple closed curves in R 3 {\textbf {R}}^{3} is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of knot types with energy less than any constant M is estimated.