Crystallisations of 2-fold branched coverings of ³

We describe the construction of a crystallisation of a 2-fold cyclic covering space of S3 branched over a link, from a bridge-presentation of the branch set. An «-dimensional ball-complex is said to be a contracted triangulation of its underlying polyhedron if it satisfies the following conditions: (i) each «-ball, considered with all its faces, is abstractly isomorphic to a closed «-simplex; (ii) the number of 0-balls (vertices) is exactly « + 1. A crystallisation of a closed, connected PL manifold M of dimension « is the edge-coloured graph, regular of degree « + 1, obtained by taking the 1-skeleton of the cellular subdivision dual to a contracted triangulation of M, and by labelling the dual of each (n — l)-simplex by the vertex it does not contain. All topological information on M is contained in such an abstract graph. A contracted triangulation turns out to be a minimal "pseudodissection" (in the sense of (HW)). The advantage of a pseudodissection is that its incidence structure may be simpler (often much simpler) than the one of a simplicial complex triangulating the same space, while the cells composing it still are simplexes. When the space is a manifold, minimality yields: (1) the existence of a "rninimal" atlas (in the sense of (PJ), and (2) the representation by a crystallisation, which, as a graph, belongs to a very circumscribed class (see (F); in dimension 3 the characteristics of this class are very easy to check). For 3-manifolds, crystallisations are not very different from Heegaard diagrams (see (P2)), with the advantage that the representation is completely graph-theoretical, the embedding into a splitting surface being possible but not necessary (also, a crystallisation embeds into three generally nonequiv- alent splitting surfaces). As a consequence, methods for finding invariants,