Hyperbolic 3-manifolds as 2-fold Coverings According to Montesinos

The Fibonacci manifolds and the ten smallest known compact orientable three-dimensional hyperbolic manifolds are obtained as twofold coverings of the three-dimensional sphere. The corresponding branch sets are described. It is shown that every manifold under consideration admits a Heegaard splitting of genus two. In this paper we consider compact orientable hyperbolic three-dimensional manifolds. We are interesting in two families of such manifolds. First of them are the Fibonacci manifolds M n , n 2, uniformized by the Fibonacci groups F(2; 2n). Manifolds M n were considered by J. Mennicke, H. Helling and A. C. Kim in 20] and 8]. These manifolds are hyperbolic for n 4. It was shown in 11] that M n can be obtained as the n-fold cyclic covering of the 3-sphere, branched over the gure-eight knot. In the previous preprint 19] we showed that for any n 2 the volume of the compact hyperbolic Fibon-acci manifold M 2n is exactly equal to the volume of the non-compact manifold S 3 n Th n , where the Turk's head link Th n is a closed 3-strings braid (1 ?1 2) n. In section 1 we will prove that the Fibonacci manifold M n , n 2, can be obtained as a twofold covering of the 3-sphere, branched over the Turk's head link Th n. In section 2 we will discuss the second family of compact 3-manifolds W(m; n; p; q) which can be obtained by (m; n) and (p; q) Dehn surgeries on two components of the Whitehead link. According to the Montesinos theorem 22], any manifold, obtained by Dehn surgeries on a strongly invertible link can be presented as a 2-fold covering of the 3-sphere, branched over some link.