Graphs encoding 3-manifolds of genus 2 (Q2468012)

In this paper, the authors introduce the so called extended Heegaard diagrams of genus \(2\). This allows to represent all 3-manifolds of genus at most two by six natural numbers, reproving a theorem of \textit{M. R. Casali} and \textit{L. Grasselli} [Discrete Math. 87, No. 1, 9--22 (1991; Zbl 0716.57001)]. This method allows to give some topological interpretation of the six natural numbers, and of the arithmetic constraints that they must satisfy (not all 6-tuples of natural numbers give a manifold). Many examples of homology 2-spheres are described.