Elastic curves and Willmore Hopf-Tori

Polynomial Killing fields and spectral curveSpectral curveClosing conditionsLet X be a periodic polynomial Killing field with period ρ, andextended frame F λ . Then the corresponding curve is closed if andonly if the monodromy M λ = F λ (ρ) satisfiesM 1 = M −1 = 1 or M 1 = M −1 = −1 .From F −1λX 0F λ = X and X 0 = X(ρ) it follows that[X 0 , M λ ] = 0 .An eigenvalue µ of M λ is a meromorphic function on Σ whose onlypoles are simple poles over the two points ∞ ± .