Knot types of generalized Kirchhoff rods

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimensional Grassmann manifold. The critical framed curves have surprisingly simple parameterizations, but they still exhibit interesting topological features. In particular, we show that for each critical energy level there is a one-parameter family of framed curves whose base curves pass through exactly two torus knot types, echoing a similar result of Ivey and Singer for classical Kirchhoff energy. In contrast to the classical theory, the generalized functional has knotted critical points which are not torus knots.