A labelled molecule is a graph G some of whose vertices and edges are equipped with labels of different kind. Labeled vertices are called atoms. There are five types of vertices:
These labels have the following meaning. Vertices of the first three types correspond to standard 3-manifolds and are considered as known atoms. Each Mobius atom is an oriented S^1-bundle over a once punctured Mobius strip. The boundary tori are equipped with coordinate systems so that the meridians are the boundary components of a section of the bundle while the longitudes are fibers. The meridians and longitudes are oriented so that the intersection numbers of the meridian with the corresponding longitudes are equal. There are eight choices of such orientations. Any two of them are related by a homeomorphism of the atom.
Triple vertices correspond to direct products N of twice punctured discs by the circle. We introduce coordinate systems on the boundary tori of N as follows: the meridians are the boundary circles of an oriented section endowed with the induced orientations. The longitudes are oriented fibers, and the intersection numbers of the meridians with the corresponding longitudes are equal. Any two such triples of coordinate systems differ by a homeomorphism of N.
Of course, solid torus vertices correspond to solid tori. Each such torus V must be labeled by an ordered pair (p,q) of coprime integers. These integers are the coordinates of the meridian of V in some coordinate system on the boundary of V. An explicit presentation of this system is not needed. We only require that the matrix label (see below) for the incoming or outgoing edge must be written with respect to the same system.
Exceptional atoms can correspond to known 3-manifolds, for example, to hyperbolic 3-manifolds contained in known tables or to complements of tabulated knots. They can also correspond to 3-manifolds which are unknown in the sense that we only know their spines.
Virtual vertices correspond to empty atoms. They are introduced to indicate the valences of other atoms and stand for their toral boundary components.
The edges of G are also of one of two types: true edges with vertices of types 1--4 and virtual edges that have a virtual vertex. Each true edge e is oriented and labeled by an order 2 matrix A with determinant +1 or -1. We can reverse the orientation of e, but then A must be replaced by its inverse. Notice that virtual edges are endowed by coordinate systems. In case of the surgery presentation, these coordinate systems are the canonical ones, otherwise they are constructed by the program according to its internal procedure (and so do not have any particular meaning). In the second case the construction of such non-canonical systems can be disabled by deselecting "Options|Always Create Coordinates On Boundary Tori".
Every labelled molecule G determines a connected 3-manifold M(G), whose boundary consists of spheres and tori. To reconstruct M(G), we replace each vertex v of type 1--4 by a copy of the corresponding manifold. The boundary tori of this manifold are parameterized by the edges that have an endpoint in v. If e is a true edge then the corresponding tori are equipped with coordinate systems, and e is labeled with a matrix. We glue together the manifolds that correspond to the endpoints of e via the homeomorphism described by the matrix. Of course, if G is disconnected, then so is the obtained 3-manifold M(G). In this case we replace it by the connected sum of all its components (since we are considering only oriented manifolds, the connected sum is well-defined).