  A physically natural potential energy for simple closed curves in $\bold R^3$
is shown to be invariant under M\"obius 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.
