2-Symmetric Transformations for 3-Manifolds of Genus 2

As previously known, all 3-manifolds of genus 2 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 2; in fact, it allows an equivalence relation to be defined 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.