Optimal planar immersions of prescribed winding number and Arnold invariants

Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter $\delta >0$ in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as $\delta \to 0$. Moreover, we show that any sequence of minimizers subconverges in $C^1$, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter $\delta$ is sufficiently small. Therefore, this limit curve serves as an “optimal” curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.