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 2 described by 6-tuples of complexity at most 21. 1 Closed 3-manifolds In this preliminary section we recall some definitions and known facts about orientable 3-manifolds without boundary. We denote the n-dimensional Euclidean space by E, the unit ball by {x ∈ E : ‖x‖ ≤ 1} by B, and the unit sphere {x ∈ E : ‖x‖ = 1} by S. We will call a space homeomorphic to B (S) a n-cell ((n− 1)-sphere). Finally, by Dn = Bn − S n−1 we denote the open ball. Definition 1 (3-manifold without boundary) [12] A topological 3-manifold is a separable metric space each of whose points has an open neighbourhood homeomorphic to E. In what follows all the considered 3-manifolds will, unless otherwise stated, assumed to be compact, connected and orientable. Each 3-manifold M gives rise to a fundamental group π1(M). The fundamental group of a 3-manifold is finitely generated. Homology group. A commutator [a, b] of elements a, b of the group G is the element [a, b] = abab. The derived subgroup G G generated by all commutators is known to be normal. The factor G/G is called an abelianisation