2-symmetric transformations for 3-manifolds of genus two

Abstract

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.