$J^+$-like Invariants under Bifurcations

TL;DR

This paper investigates how specific topological invariants of smooth closed curves in the plane change during bifurcations created by multiple traversals and perturbations, enhancing understanding of curve invariants under complex deformations.

Contribution

It introduces a detailed analysis of the behavior of $J^+$, $J^-$, $ ext{J}_1$, and $ ext{J}_2$ invariants under $k$-bifurcations, extending prior invariance results to more complex curve transformations.

Findings

Derived formulas for invariant changes under $k$-bifurcations

Identified conditions where invariants remain stable or change

Provided insights into the topological behavior of immersed curves

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.