Interlacement of Double Curves of Immersed Spheres

Abstract

We characterize those unions of embedded disjoint circles in the sphere $$S^2$$S2 which can be the multiple point set of a generic immersion of $$S^2$$S2 into $$\mathbb {R}^3$$R3 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 (Manuscr Math 113(2):239---250, 2004) and we further generalize the ideas of de Fraysseix and de Mendez (Discrete Comput Geom 22:287---295, 1999), which leads us to directed interlacement graphs of paired trees and their local complementation.