Isomorphisms and homeomorphisms of a class of graphs and spaces (Q1849456)

Lins-Mandel manifolds are a particular class of closed orientable 3-manifolds, which are represented by 4-coloured graphs depending on four parameters, called Lins-Mandel gems [\textit{S. Lins} and \textit{A. Mandel}, Discrete Math. 57, 261-284 (1985; Zbl 0585.57009); \textit{A. Cavicchioli}, ibid., 17-37 (1985; Zbl 0579.57008); \textit{M. R. Casali} and \textit{L. Grasselli}, ibid. 87, No.~1, 9-22 (1991; Zbl 0716.57001); \textit{M. Mulazzani}, ibid. 140, No.~1-3, 107-118 (1995; Zbl 0841.57026)]; as far as the topological point of view is concerned, it is known that each Lins-Mandel manifold is a cyclic covering of \(\mathbb{S}^3\) branched over a two-bridge knot or link [\textit{M. Mulazzani}, J. Knot Theory Ramifications 5, No.~2, 239-263 (1996; Zbl 0858.57004)]. In the present paper, the authors completely solve the isomorphism problem for the whole class of Lins-Mandel gems. Moreover, by applying to Lins-Mandel manifolds suitable results on branched cyclic coverings of two-bridge hyperbolic links (obtained in [\textit{B. Zimmermann}, Topology Appl. 65, No.~3, 287-294 (1995; Zbl 0848.57012); \textit{M. Sakuma}, Kobe J. Math. 7, No.~2, 167-190 (1990; Zbl 0727.57007)]), the isomorphism conditions for Lins-Mandel gems are proved to be equivalent -- in a wide subset of interesting cases -- to the homeomorphism conditions for the represented 3-manifolds.