PHYSICALLY-BASED STOCHASTIC SIMPLIFICATION OF MATHEMATICAL KNOTS

The article describes a tool for simplification and analysis of tangle d configurations of mathematical knots. The proposed method addresses optimization issues common in energy-based approaches to knot classifi cation. in this class of methods, an initially tangled elastic rope is ''charged'' with an electrostalic-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 typ e. We propose a physically-based model to implicitly guard against iso topy violation during such evolution and suggest that a robust stochas tic optimization procedure, simulated annealing, be used for the purpo se of identifying the globally optimal solution. Because neither of th ese techniques depends on the properties of the energy function being optimized, our method is of general applicability, even though we appl ied it to a specific potential here. The method has successfully analy zed several complex tangles and is applicable to simplifying a large c lass of knots and links. Our work also shows that energy-based techniq ues will not necessarily terminate in a unique configuration, thus we empirically refute a prior conjecture that one of the commonly used en ergy functions (Simon's) is unimodal. Based on these results we also c ompare techniques that rely on geometric energy optimization to conven tional algebraic methods with regards to their classification power.