The only conditions that I know of under which it's absolutely known that gradient flow will converge to a minimum energy state is when the initial configuration is planar. Zheng-Xu He proved this for his renormalized $1/r^2$ potential 'Mobius' energy in *The Euler-Lagrange equation and heat flow for the Möbius energy* (CPAM 53, 2000). We proved a similar theorem for 'repulsive' energies on planar polygons in *An energy-driven approach to linkage unfolding* (SOCG 2004).

For a nonplanar configuration, you're right to think that there's a general theorem that ``simulated annealing has a positive probability of converging to the global minimum" which applies to these problems, but it's ineffective in practice. The configuration space is in practice very high dimensional (3 x number of vertices) and reducing an energy functional such as ropelength generally requires a coordinated global motion of these vertices, so it's quite rare to generate such a move randomly. AFAIK, nobody knows how to estimate the probability of convergence under these circumstances, so there's no hard information on how long to run an annealer.

FWIW, *RidgeRunner* really isn't an annealer: it generates 'coordinated motions' for tightening deterministically using a linear algebra algorithm to deflect the gradient of length for a polygon into a ropelength-decreasing direction using the active distance and curvature constraints. The algorithm is described in *Knot Tightening By Constrained Gradient Descent* (Experimental Math, 2011).

You're welcome to play with *RidgeRunner* yourself if you'd like to try tightening: the software can be downloaded at link text.