Crystallizations of generalized Neuwirth manifolds

It is well known that every Heegaard diagram canonically induces a (balanced) presentation of the fundamental group of the represented 3-manifold. Given a group presentation P, let M(P) denote the class of all 3-manifolds which admit a Heegaard diagram inducing P; if M(P) is nonvoid, then P is said to be a geometric group presentation. The present paper makes use of the manifold representation theory by edge-coloured graphs in order to derive a new algorithm, which efficiently tests the geometricity of the presentation P and produces crystallizations of all 3-manifolds of M(P). Moreover, this new approach allows to construct crystallizations of a wide class of Seifert fibered 3-manifolds, which generalizes, in a natural way, the so-called Neuwirth manifolds (introduced by Neuwirth as an application of his own geometricity algorithm).