Fundamental groups of branched covering spaces ofS 3

Summary Given a knotK⊂S ³, it is known a standard method for constructing a 4-coloured graph representing the closed orientable 3-manifoldM=M(K, d, ω) which is thed-fold covering space ofS ³ branched overK and associated to the transitived-representation ω of the knot group. In this paper we obtain a presentation of the fundamental group ofM, directly from the Wirtinger presentation of the knot group and from the transitived-representation ω.

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Riassunto Dato un nodoK⊂S ³, è noto un metodo standard per costruire un grafo 4-colorato rappresentante la 3-varietà chiusa ed orientabileM=M(K, d, ω) che è lo spazio di rivestimento diS ³ ramificato suK ed associato allad-rappresentazione transitiva ω del gruppo del nodo. In questo articolo si ottiene una presentazione del gruppo fondamentale diM, direttamente dalla presentazione di Wirtinger del gruppo del nodo e dallad-rappresentazione transitiva ω.

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Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy) and financially supported by M.P.I. (project ?Geometria delle Varietà differenziabili?).     

The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

The homology of irregular dihedral branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

A note on monotone covering properties

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On \pi-hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation if is homeomorphic to . It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related. Received: 3 August 1998 / in final form: 17 June 1999     

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