Closed planar curves without inflections

TL;DR

This paper introduces a computable topological invariant for generic closed planar curves, providing a lower bound for inflection points and classifying locally convex curves with few crossings, linking it to double tangents.

Contribution

It defines a new invariant for classifying and analyzing closed planar curves, especially those without inflections, and explores its relationship with double tangents.

Findings

Invariant effectively bounds inflection points.

Classified locally convex curves with up to five crossings.

Linked the invariant to the number of double tangents.

Abstract

We define a computable topological invariant $\mu(\gamma)$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $\mu(\gamma)$ on a given $\gamma$.