Abstract

In the present paper links and knots are considered as singular subsets of geometric cone-manifolds with the three-sphere as an underlying space. Trigonometrical identities between lengths of singular components and cone angles for the figure eight knot, Whitehead link and Borromean rings are obtained. This gives a possibility to express the lengths in terms of cone angles. Then the Schläfli formula applies to find explicit formulae for hyperbolic and spherical volumes of these cone-manifolds.

Introduction

In 1975 R. Riley [Ril79] found examples of hyperbolic structures on some knot and link complements in the three-sphere. Later, in the spring of 1977, W. P. Thurston announced an existence theorem for Riemannian metrics of constant negative curvature on 3-manifolds. In particular, it turned out that knot complement of a simple knot (excepting torical and satellite) admits a hyperbolic structure. This fact allowed to consider knot theory as a part of geometry and Kleinian group theory. Starting from Alexander's works polynomial invariants became a convenient instrument for knot investigation. A lot of different kinds of such polynomials were investigated in the last twenty years. Among these there are Jones-, Kaufmann-, HOMFLY-, A–polynomials and others ([Kau88], [CCG+94], [HLM95a]). This relates the knot theory with algebra and algebraic geometry. Algebraic technique is used to find the most important geometrical characteristics of knots such as a volume, length of shortest geodesics and others.