Equivalent elastica knots

Abstract

The problem of an elastica knot in three-dimensional space is solved explicitly by expressing the Frenet-Serret curvature and torsion of the knot in terms of the Weierstrass and Jacobi elliptic functions. This solution is obtained by variational methods and is derived by minimizing of the squared-curvature energy integral. In the present work, an equivalency is established between pairs of Jacobi elliptic solutions that are described by the same values for curvature and torsion functionals.