Right. 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 where |x-y| is the distance through Euclidean space, d(x,y) is the distance along the knot, dx, dy elements of arclength along the knot. (The abuse is just that the pieces of the integrand go to infinity at the diagonal x = y. But fortunately, if you avoid integrating over an epsilon-neighborhood of the diagonal, the integral converges as epsilon approaches 0.) The belief that sliding down the gradient of E(K) will never get stuck was already believed by at least 1995. --Dan On 2013-12-31, at 7:05 AM, Veit Elser wrote:
The spookiest untangling algorithm I've seen is to flow along the downhill gradient of a 1/r^2 repulsive potential (between all pairs of line elements). This energy is scale invariant, but less obviously, Moebius invariant. John Sullivan showed me movies of some very complex tangles transforming effortlessly into a nice round unknot. Last time I spoke with him no one had found an example of a non-trivial critical point of this energy, i.e. a mechanism for the algorithm to get stuck. The motion is global in character, with tightly tangled regions spontaneously expanding and then unraveling.