Geodesically complete Riemannian metrics on the space of knots

May 23, 2023
Self-avoiding energies were originally constructed to simplify knots and links in R^3. The driving idea was to design barrier functions for the feasible set, the set of curves of prescribed isotopy class. Such functions must blow up whenever a path of curves tries to escape the feasible set. This singular behavior makes it challenging to perform numerical optimization for self-avoiding energies, in particular, when ""close"" to the boundary of the feasible set. Motivated by the Riemannian metric in the Poincaré model of hyperbolic space, my collaborators and I constructed Riemannian metrics that provide nice preconditioning for the Möbius energy (with Philipp Reiter) and for tangent-point energies (with Keenan Crane). These metrics work extremely well in numerical experiments. In particular, the elementary gradient schemes that we employ require rarely any line search in the form of collision detection. That is, a step of finite, reasonable size in the knot space (almost) never escapes from the feasible set. That lead us to the conjecture that these metrics (or some mild modifications) must be geodesically complete. In this talk I will present ongoing work with Elias Döhrer and Philipp Reiter. After explaining the notion of geodesic completeness and its applications, I will introduce a certain class of Riemannian metrics on the space of knots. These are closely related to tangent-point energies whose energy spaces are Hilbert spaces. Finally, I will sketch a proof for geodesic completeness.