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

It is known that every closed compact orientable 3-manifold M can be represented by a 4-edge-coloured 4-valent graph called a crystallisation of M. Casali and Grasselli proved that 3-manifolds of Heegaard genus g can be represented by crystallisations with a very simple structure which can be described by a 2(g+1)-tuple of non-negative integers. The sum of first g+1 integers is called complexity of the admissible 2(g+1)-tuple. If c is the complexity then the number of vertices of the associated graph is 2c. In the present paper we describe all prime 3-manifolds of Heegaard genus two described by 6-tuples of complexity at most 21.

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