Correction: Veit informs me that it was Jun O'Hara who first invented this energy, and Freedman who first noticed that it is Möbius-invariant.* ----------- --Dan _____________________________ * In a paper with He and Wang P.S. Just to be clear, Möbius-invariant here means conformally-invariant: Let f: S^3 -> S^3 be any angle-preserving transformation. All of them form the conformal (or Möbius) group of S^3. By stereographic projection S: S^n - {*} -> R^n, where * = (0,0,0,1), (with inverse Sinv), f -- or more precisely S o f o Sinv -- is *almost* a homeomorphism of R^n. Since S preserves angles on S^3, so does S o f o Sinv on R^3, where it is defined. So for a knot K in R^n, as long as f(Sinv(K)) does not contain *, we can apply S to it to get K' := S(f(Sinv(K)), another knot in R^3. The statement of conformal or Möbius invariance of E(K) means that for any such new knot, we have E(K') = E(K). The conformal transformations "of R^3 to itself" are generated by translations, rotations, uniform scaling, and inversion in a sphere. (But the last one in undefined at the center of the sphere, taking that point to "infinity".) I wrote:
The energy that I think Sullivan -- and most researchers who talk of Möbius-invariant energy -- work with this one usually attributed to Mike Freedman: It's (by slight abuse of notation), for a smoothly embedded knot K, this double integral over the cartesian square of the knot:
E(K) := Int Int over KxK of (1/|x-y|^2 - 1/(d(x,y)^2) dx dy