Loki: Software for Computing Cut Loci

Abstract

The first quantitatively correct pictorial atlas of the cut locus of a nontrivially deformed standard torus in $\mathbb{R}^3$ given a non-symmetrically placed starting point is presented along with a description of the software tool Loki used to generate it. Loki can compute the cut locus from a point on a genus-1 two-dimensional Riemannian manifold defined either by a parametrization or its metric, these to be given in closed form. The algorithm computes a piecewise polynomial approximation to the exponential map and inverts this numerically, thus correctly taking into account the global nature of the problem. As an example of its use in motivating and guiding traditional mathematical research, we provide a preliminary conjecture based upon the output of this software and both a counterexample and a proof motivated by the conjecture.