Knot types of generalized Kirchhoff rods

TL;DR

This paper introduces a generalized energy functional for elastic rods that accounts for axial stretch and inflation, providing explicit parameterizations of all periodic critical curves and revealing new knotted configurations beyond classical torus knots.

Contribution

It extends classical Kirchhoff energy to a broader class of rods, deriving explicit formulas for critical points and uncovering novel knotted solutions.

Findings

Explicit parameterizations for all periodic critical framed curves.

Existence of a one-parameter family passing through two torus knot types.

Identification of knotted critical points that are not torus knots.

Abstract

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.