Narrow Results

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the volumes of the cyclic coverings over the hyperbolic twist knots.

Let $C(2n,3)$ be the family of two bridge knots of slope $(4n+1)/(6n+1)$. We calculate the volumes of the $C(2n,3)$ cone-manifolds using the Schläfli formula. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano and Montesinos-Amilibia and extend the Ham, Mednykh and Petrov’s methods. As an application, we give the volumes of the cyclic coverings over those knots. For the fundamental group of $C(2n,3)$, we take and tailor Hoste and Shanahan’s. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert’s canonical two-bridge diagram or not.

In the present paper links and knots are investigated as a singular set of geometric cone–manifolds with the three-sphere as underlying space. Trigonometric identities between lengths of singular components and cone angles of these cone–manifolds (Sine, Cosine, and Tangent rules) are obtained. Geometrical inequalities between volumes and singular geodesic lengths of the cone–manifolds are also given. They can be considered as a sort of isoperimetric inequalities well-known for convex polyhedra.

Denote by W(m, n) the hyperbolic cone-manifold whose underlying space is the 3-sphere and singular geodesics are formed by two components of the Whitehead link with cone angles 2π/m and 2π/n. The aim of the paper is to establish the Tangent and Sine Rules relating the complex lengthes of the singular geodesics and the cone angles of W(m, n). An explicit upper bound for the real length of the singular geodesic is also given.