  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.
