M ar 2 00 0 2 - SYMMETRIC TRANSFORMATIONS FOR 3 - MANIFOLDS OF GENUS TWO

As previously known, all 3-manifolds of genus two can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an “elementary transformation” on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifolds of genus two; in fact, it allows to define an equivalence relation on “admissible” 6-tuples so that equivalent 6-tuples represent the same manifold. Different equivalence classes can represent the same manifold; however, equivalence classes “almost always” contain infinitely many 6-tuples. Finally, minimal representatives of the equivalence classes are described.