M\"obius invariant metrics on the space of knots

We give a condition for a function to produce a M"obius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of M"obius invariant knot energies can produce M"obius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of M"obius geometry.