Untangling knots by stochastic energy optimization

A method for visualizing unknottedness of mathematical knots via energy optimization with simulated annealing is presented. In this method a potential field is formed around a tangled rope that causes it to self-repel. By allowing the rope to evolve in this field in search of an energy minimizing configuration we can determine the knot type of the initial configuration. In particular, it is natural to conjecture that if such a "charged rope" was not initially knotted, it will reach its minimal potential in a circular configuration, given a suitable energy functional. Because situations potentially arise in which the functional may not be strictly unimodal, we suggest it to be advantageous to use a robust stochastic optimization technique (simulated annealing), rather than a deterministic hill climber common in physically based approaches, to make sure that the evolving rope does not settle in a suboptimal configuration. The same method is applicable to simplifying arbitrary knots and links and for establishing knot equivalence. Aside from its theoretical appeal, the method promises to solve practical problems common in genetic research and polymer design.