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We characterize those unions of embedded disjoint circles in the sphere (Formula presented.) which can be the multiple point set of a generic immersion of (Formula presented.) into (Formula presented.) 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. © 2016, Springer Science+Business Media New York.
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