I will report on advances in the regularity theory for minimizers and critical points of a class of knot energies defined by Jun O'Hara. When parametrized by arclength the tangent field of these knots are critical points of a W^{1/p,p}-type energy, and we employ arguments from the regularity theory of W^{1/p,p}-harmonic maps into the sphere.

Joint work with S. Blatt, Ph. Reiter.