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. -Veit On Dec 30, 2013, at 6:15 PM, Cris Moore <moore@santafe.edu> wrote:
beautiful movies!
I have heard that making the units of string repel each other, while keeping the arc length constant, seems to be a good heuristic for unknotting unknots. However, since telling whether a knot is an unknot or not is believed to be computationally hard (even showing it's in NP is pretty challenging) such a heuristic shouldn't work in all cases --- there should exist knots where it gets stuck.
A similar issue arises in numerical general relativity: it shouldn't be easy to tell whether two 4-manifolds are equivalent (it's undecidable because of the word problem!) but there are no known examples where numerical algorithms that try to tell whether a 4-manifold is just a sphere get stuck for a long time.
Does anyone know of references on either of these?
Cris