On double points of immersions of spheres (Q1434140)

This paper studies immersions of the sphere \(S^2\) in \(\mathbb{R}^3\) which do not contain triple points and the double points are either {intersection points} or {touching points}. An immersion provides a finite collection of disjoint circles \(S^1\) in the domain \(S^2\), to which one can associate a graph which is a tree. The author remarks that not all trees are realizable. The purpose of the paper is to give a purely combinatorial characterization of the trees which are realizable. He uses surgeries to obtain the result. The paper is well written including the figures which are very helpful.