Introduction to Arnold's $J^+$-Invariant

TL;DR

This paper introduces Arnold's $J^+$-invariant for planar immersions, explaining computation methods and providing accessible examples and exercises suitable for undergraduates.

Contribution

It offers an accessible introduction to Arnold's $J^+$-invariant, including computation techniques like Viro's sum, with educational content for beginners.

Findings

Provides a clear explanation of Arnold's $J^+$-invariant

Includes practical computation methods such as Viro's sum

Offers examples and exercises for learning

Abstract

We explore Arnold's $J^+$-invariant of immersions -- planar smooth closed curves with non-vanishing derivative, at most double points and only transverse intersections -- and computation methods like Viro's sum, among others. Only basic undergraduate mathematics is needed to understand the contents of this introductory paper and everything we need that is above that is recalled or introduced. Examples, exercises and solutions are included for practice.