Physically-based stochastic simplification of mathematical knots

摘要

The article describes a tool for simplification and analysis ofntangled configurations of mathematical knots. The proposed methodnaddresses optimization issues common in energy based approaches to knotnclassification. In this class of methods, an initially tangled elasticnrope is “charged” with an electrostatic like field whichncauses it to self repel, prompting it to evolve into a mechanicallynstable configuration. This configuration is believed to bencharacteristic for its knot type. We propose a physically based model tonimplicitly guard against isotopy violation during such evolution andnsuggest that a robust stochastic optimization procedure, simulatednannealing, be used for the purpose of identifying the globally optimalnsolution. Because neither of these techniques depends on the propertiesnof the energy function being optimized, our method is of generalnapplicability, even though we applied it to a specific potential here.nThe method has successfully analyzed several complex tangles and isnapplicable to simplifying a large class of knots and links. Our worknalso shows that energy based techniques will not necessarily terminatenin a unique configuration, thus we empirically refute a prior conjecturenthat one of the commonly used energy functions (J. Simon, 1994) isnunimodal. Based on these results we also compare techniques that rely onngeometric energy optimization to conventional algebraic methods withnregards to their classification power