  We develop a regularity theory for extremal knots of scale invariant knot
energies defined by J. O'hara in 1991. This class contains as a special case
the M\"obius energy.
  For the M\"obius energy, due to the celebrated work of Freedman, He, and
Wang, we have a relatively good understanding. Their approch is crucially based
on the invariance of the M\"obius energy under M\"obius transforms, which fails
for all the other O'hara energies.
  We overcome this difficulty by re-interpreting the scale invariant O'hara
knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent
of the knot parametrization. This allows us to draw a connection to the theory
of (fractional) harmonic maps into spheres. Using this connection we are able
to adapt the regularity theory for degenerate fractional harmonic maps in the
critical dimension to prove regularity for minimizers and critical knots of the
scale-invariant O'hara knot energies.
