A classical problem in the geometric calculus of variations is the problem of finding particularly “nice” representatives within a class of geometric objects. These representatives are usually achieved by finding critical points of an energy functional acting on these objects.

Quite surprisingly, even in very classical situations, these questions are intrinsically related to fractional order energies and fractional objects.

We will have a look at knot energies, which are related to fractional versions of geodesics in the 2-sphere. Then we will have a look at how this relation helps us to answer questions about knot energies.