On the Wu invariants for immersions of a graph into the plane

TL;DR

This paper explicitly calculates Wu invariants for graph immersions into the plane and classifies all such immersions up to regular homotopy, extending classical results for plane curves.

Contribution

It provides a comprehensive method to compute Wu invariants for graph immersions and classifies them up to regular homotopy, generalizing the rotation number classification.

Findings

Explicit calculation method for Wu invariants

Classification of graph immersions up to regular homotopy

Extension of classical plane curve results

Abstract

We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization of the fact that two plane curves are regularly homotopic if and only if they have the same rotation number.