Three-manifolds with Heegaard genus at most two represented by crystallisations with at most 42 vertices Original Research Article

It is known that every closed compact orientable 3-manifold image can be represented by a 4-edge-coloured 4-valent graph called a crystallisation of image. Casali and Grasselli proved that 3-manifolds of Heegaard genus image can be represented by crystallisations with a very simple structure which can be described by a image-tuple of non-negative integers. The sum of first image integers is called complexity of the admissible image-tuple. If image is the complexity then the number of vertices of the associated graph is image. In the present paper we describe all prime 3-manifolds of Heegaard genus two described by 6-tuples of complexity at most 21.