On lacets and their manifolds

Abstract

By topological lacet we mean the embedding, with self-intersections, of a closed curve in a 2-manifold, such that (1) all self-intersections are simple double points at which the curve crosses itself, and (2) the complement of the curve in the 2-manifold is a 2-colorable family of discs. Such embeddings are characterized, up to homotopy, by a combinatorial lacet, that is, by a double occurrence word Δ on an alphabet P and a bipartition (K,L) of the set P, P representing the set of self-intersections of the curve, Δ their sequence in one complete run along the curve, and K,L the two types of reentry possible at a crossing. Passing by the construction of a map associated to the lacet, we show that every combinatorial lacet is representable on a 2-manifold. Beginning with a problem on curves in the plane posed and partially solved by Gauss (1840), and introducing certain linear transformations of vector spaces over GF(2) , depending on Δ and (K,L), Rosenstiehl [C. R. Acad. Sci. Paris Ser. A 283 (1976) 551] and Lins et al. [Aequationes Math. 33 (1987) 81] characterized combinatorial lacets representable in the Euclidean plane and projective plane, respectively, by a simple property of the enlacement graph of pairs of letters in Δ. We prove here that a characterisation of the same sort extends naturally to the case of arbitrary 2-manifolds. In particular, we present procedures for determining if a combinatorial lacet is representable on the torus, or on the Klein bottle.