$J^+$-like Invariants under Bifurcations

Abstract: We explore how the invariants $J^+$, $J^-$, $\mathcal{J}_1$ and $\mathcal{J}_2$ of immersions -- generic (at most double points and only transverse intersections) planar smooth closed curves with non-vanishing derivative -- change under $k$-bifurcations ($k \ge 2 \in \mathbb{N}$), which are constructed by running $k$ times through an immersion and then perturbing it to be generic.