How nice are critical knots of knot energies? We already know that critical points of the so-called Möbius energy with merely bounded energy are smooth. This leads to the question whether smooth critical points of the Möbius energy are also real analytic.
In this talk, we give a quick introduction to geometric knot theory with focus on the Möbius energy and present an essential technique, with which the open question on the analyticity of smooth critical points of the Möbius energy was solved. To the best of the authors' knowledge, this is one of the first analyticity results in the context of non-local differential equations.