# On lacets' and their manifolds

## ABSTRACT

We study the class of embeddings of a simple closed curve in closed 2-manifolds such that {\parindent=24pt\parskip=3pt \item{(1)} there are only transversal double points, \item{(2)} each residual region of the manifold is a disc, and \item{(3)} this set of discs is two-colorable, thus furnishing the vertices and faces of a combinatorial map (carte'). } Such embeddings are characterized up to homotopy by two combinatorial structures on the set $E$ of crossing points: {\parindent=24pt\parskip=3pt \item{(1)} The double occurrence word $\Delta$, or `combinatorial lacet', giving the order of crossing points along the curve, and \item{(2)} a bipartition $(K,L)$ of the set $E$, classifying the type of reentry at each crossing.} \vskip0pt This combinatorial information gives rise to a unique map having the lacet as its diagonal. Given a combinatorial lacet, on what 2-manifolds can it be represented? Starting from a problem and partial response by Gauss (1840), and using transformations of vector spaces over $GF(2)$, Rosenstiehl (1976) and Lins, Richter \& Shank (1987) characterized lacets representable in the Euclidean and projective planes, respectively. We show how these characterizations extend naturally to the case of arbitrary 2-manifolds, and give, in particular, simple procedures for determining whether a lacet is representable on the torus or on the Klein bottle. This is joint work with Pierre Rosenstiehl.

Speaker's Contact Info: Henry.Crapo(at-sign)inria.fr