Physically-based stochastic simplification of mathematical knots

Abstract

The article describes a tool for simplification and analysis of tangled configurations of mathematical knots. The proposed method addresses optimization issues common in energy based approaches to knot classification. In this class of methods, an initially tangled elastic rope is charged with an electrostatic like field which causes it to self repel, prompting it to evolve into a mechanically stable configuration. This configuration is believed to be characteristic for its knot type. We propose a physically based model to implicitly guard against isotopy violation during such evolution and suggest that a robust stochastic optimization procedure, simulated annealing, be used for the purpose of identifying the globally optimal solution. Because neither of these techniques depends on the properties of the energy function being optimized, our method is of general applicability, even though we applied it to a specific potential here. The method has successfully analyzed several complex tangles and is applicable to simplifying a large class of knots and links. Our work also shows that energy based techniques will not necessarily terminate in a unique configuration, thus we empirically refute a prior conjecture that one of the commonly used energy functions (J. Simon, 1994) is unimodal. Based on these results we also compare techniques that rely on geometric energy optimization to conventional algebraic methods with regards to their classification power.