Elastic curves and Willmore Hopf-Tori

Whitham deformationExampleThe branchpoints of the spectral curve of an elastic curve are theroots of the real even quartic polynomialm 4 + m 2 λ 2 + λ 4 .With discriminant ∆ = 1 4 m 2 − m 4 , away from degeneratesituations, there are two caseswave-like if ∆ < 0: distinct branchpoints ±p 1 , ±p 2 with p 2 = ¯p 1orbit-like if ∆ > 0: distinct branchpoints on iR × .In the case ∆ = 0 the spectral curve is singular.Figure: Spectral curves of wave-like and orbit-like elastic curves.