We give a constructive proof of a Theorem of Izmestiev and Joswig. Namely, given $(L,\omega)$ where $L$ is a link in $S^{3}$ and $\omega$ a simple (not necessarily transitive) representation of $\pi_{1}(S^{3}\backslash L)$ onto the symmetric group $\Sigma_{4}$ of four elements $\{1,2,3,4\}$ we construct a triangulation of $S^{3}$ giving rise to $(L,\omega)$ in a natural way.

Arithmetic and non arithmetic hyperbolic 3-manifolds having infinitely many fibrations over $S^{1}$ of arbitrary high genus are constructed. A concrete example is studied in detail.

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. The paper explains a method of computing the excellent component of the character variety of tunnel number 1 knots.

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. In this paper we explain a method to compute the excellent component of the character variety of periodic knots. We apply the method to those knots obtained as the preimage of one component of a 2-bridge link by a cyclic covering of S3 branched on the other component. We call these knots periodic knots with rational quotient. Among this class of knots are the ‘Turk's head knots’. Finally we give some invariants deduced from the excellent component of the character curve, such as the h-polynomial and the limit of hyperbolicity for all the periodic knots with rational quotient, up to 10 crossings, which are not 2-bridge or toroidal.

One important invariant of a closed Riemannian 3-manifold is the Chern–Simons invariant [1]. The concept was generalized to hyperbolic 3-manifolds with cusps in [11], and to geometric (spherical, euclidean or hyperbolic) 3-orbifolds, as particular cases of geometric cone-manifolds, in [7]. In this paper, we study the behaviour of this generalized invariant under change of orientation, and we give a method to compute it for hyperbolic 3-manifolds using virtually regular coverings [10]. We confine ourselves to virtually regular coverings because a covering of a geometric orbifold is a geometric manifold if and only if the covering is a virtually regular covering of the underlying space of the orbifold, branched over the singular locus. Therefore our work is the most general for the applications in mind; namely, computing volumes and Chern–Simons invariants of hyperbolic manifolds, using the computations for cone-manifolds for which a convenient Schläfli formula holds (see [7]). Among other results, we prove that every hyperbolic manifold obtained as a virtually regular covering of a figure-eight knot hyperbolic orbifold has rational Chern–Simons invariant. We give explicit examples with computations of volumes and Chern–Simons invariants for some hyperbolic 3-manifolds, to show the efficiency of our method. We also give examples of different hyperbolic manifolds with the same volume, whose Chern–Simons invariants (mod ½) differ by a rational number, as well as pairs of different hyperbolic manifolds with the same volume and the same Chern–Simons invariant (mod ½). (Examples of this type were also obtained in [12] and [9], but using mutation and surgery techniques, respectively, instead of coverings as we do here.)