Tetrahedron manifolds via coloured graphs (Q1379830)

The authors study 3-manifolds of the title previously shown by \textit{E. Molnár} in [Note Mat. 10, No. 2, 335-346 (1990; Zbl 0770.57006)] to be Seifert fiber spaces having fundamental group \(G_{mn}\) with presentation \(P= (x,y\mid x^{m-1} y^{-1} x^{-1} y^{-1}, y^{n-1} x^{-1} y^{-1} x^{-1})\). The paper examines these manifolds from the point-of-view of edge-colored graphs. They show \(P\) is geometric and they reproduce Molnar's result, proving directly that these manifolds are branched 2-sheeted coverings of \(S^3\) with branching set certain links \(L_{mn}\).