# Interlacement of double curves of immersed spheres

## TL;DR

This paper characterizes unions of disjoint circles in a 2-sphere that can form the multiple point set of a generic immersion into 3-space, using interlacement and graph-theoretic concepts, extending classical Gauss code characterizations.

## Contribution

It provides a higher-dimensional analogue of Gauss code characterization for immersed spheres, employing interlacement graphs and local complementation techniques.

## Key findings

- Characterization of multiple point sets via interlacement
- Extension of Rosenstiehl's Gauss code characterization
- Use of directed interlacement graphs and local complementation

## Abstract

We characterize those unions of embedded disjoint circles in the 2-sphere which can be the multiple point set of a generic immersion of the 2-sphere into 3-dimensional space in terms of the interlacement of the given circles. Our result is the one higher dimensional analogue of Rosenstiehl's characterization of words being Gauss codes of self-crossing plane curves. Our proof uses a result of Lippner and we further generalize the ideas of Fraysseix and Ossona de Mendez, which leads us to directed interlacement graphs of paired trees and their local complementation.