Allen Hatcher, in a preprint "Topological Moduli Spaces of Knots," subsequent to his proof of Smale's conjecture, remarks that one might expect there to be a nice energy function on the space of unknots (or, for that matter, on the space of knots of any given isotopy type) with the property that gradient flow in the direction of decreased energy always terminates at a standard form for the knot (or some very low-dimensional space of standard forms.) So you could certainly ask whether one would expect the complexity of a given unknot, in whatever sense this is meant, to be monotone decreasing as you flow it towards a nice round circle.
Greg Buck gave an interesting talk here at UW where he presented a candidate for such a function, and showed several cool movies (not available) of crimped, tangled unknots resolving to round circles. Buck said that he'd never observed the flow getting caught at a local minimum. (I didn't ask him whether the knot ever "gets more tangled on its way to being less tangled," but one could define this problem away by using his energy as the definition of tangledness....!)
