Elastic curves and Willmore Hopf-Tori

Whitham deformationExampleCirclesThe simplest polynomial Killing field is( )iλ κX =−κ −iλwith κ ∈ R constant. The corresponding curves are circles with(geodesic) curvature κ, and the branchpoints of the spectral curveare located at ±iκ. Since log µ = πi√λ 2 + κ 2 , doublepoints area ∈ R with a 2 + κ 2 ∈ Z 2 .Figure:The spectral curve of a bifurcating circle.