Optimal planar immersions of prescribed winding number and Arnold invariants

TL;DR

This paper investigates the minimization of a tangent-point energy among planar curves with fixed topological invariants, proving existence, convergence, and optimality of such energy-minimizing immersions.

Contribution

It introduces a variational framework for Arnold invariants, establishing existence and convergence of energy minimizers with prescribed topological properties.

Findings

Existence of energy minimizers for each truncation parameter.

Gamma convergence of truncated energies to a renormalized limit.

Limit curves preserve topological invariants and have right-angle self-intersections.

Abstract

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.