Equivalent elastica knots

TL;DR

This paper provides an explicit solution for elastica knots in 3D space using elliptic functions, derived through variational methods, and establishes an equivalence between different elliptic solutions based on curvature and torsion.

Contribution

It introduces a novel explicit solution for elastica knots in 3D space using elliptic functions and establishes an equivalence between solutions with identical curvature and torsion.

Findings

Explicit solutions expressed via Weierstrass and Jacobi elliptic functions.

Establishment of equivalency between elliptic solutions with same curvature and torsion.

Solution derived through variational minimization of squared-curvature energy.

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.