Narrow Results

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.