Research Announcement on the “Energy” of Knots

Abstract

It is a natural idea that a loop of string might somehow be charged and placed in a bottle of honey, there to assume its least (potential) energy or “best” shape. In the late 1970’s Alan Hatcher announced a remarkable geometric theorem (Hatcher, 1983) (the “Smale conjecture”: Diff(S 3) ≃ O(4)). Among its many consequences is the fact that the space of all smooth un-knots — that is, simple closed curves in three space (R 3) which can be deformed to a round circle — actually can be continuously deformed to the subspace of round circles in three space. This has heightened interests in the honey jar experiment. For the theorem tells us that there is some method — continuous in initial conditions — which untangles arbitrary unknots and makes them round. The key point is continuity; there should be no intelligent intervention. No choices should be made. A person untangling string makes lots of judgements but perhaps the charged unknotted loop in honey would always make its way to a round circle without incident. Perhaps not. Computer experiments may be able to suggest the correct answer to this interesting mathematical problem. Actually the Newtonian potential 1/r is too weak at small scales to create an infinite energy barrier to crossings. This requires at least the potential described 1/r2 below. Whether or not weaker potentials 1/rp, P < 2 might yet determine gradient flows which respect topology (i.e., admit no crossings) is an open question.