An Invariant for Triple-Point-Free Immersed Spheres

TL;DR

This paper introduces a new invariant for triple-point-free immersed spheres in 3D space, revealing infinitely many regular homotopy classes and providing a combinatorial representation of double points.

Contribution

It defines an invariant taking values in l^1(Z) that classifies triple-point-free immersions and describes their homotopy classes using a directed tree structure.

Findings

The invariant distinguishes infinitely many homotopy classes.

Many sphere pairs cannot be connected without passing through triple points.

A combinatorial model using directed trees represents double points.

Abstract

We define an invariant of triple-point-free immersions of $2$-spheres into Euclidean $3$-space, taking values in $l^1(\mathbb{Z})$. It remains unchanged under regular homotopies through such immersions. An explicit description of its image shows that the space of triple-point-free immersed spheres has infinitely many regular homotopy classes. Consequently, many pairs of immersed spheres can only be connected by regular homotopies that pass through triple points. We represent the double points of a triple-point-free immersed sphere using a directed tree, equipped with a pair relation on the edges and an integer-valued function on the vertices. The invariant depends on this function and on the vertex indegrees.