Book contents
- Frontmatter
- Contents
- Preface
- Part I Hyperbolic 3-manifolds
- Combinatorial and geometrical aspects of hyperbolic 3-manifolds
- Harmonic deformations of hyperbolic 3-manifolds
- Cone-manifolds and the density conjecture
- Les géodésiques fermées d'une variété hyperbolique en tant que nœuds
- Ending laminations in the Masur domain
- Quasi-arcs in the limit set of a singly degenerate group with bounded geometry
- On hyperbolic and spherical volumes for knot and link cone-manifolds
- Remarks on the curve complex: classification of surface homeomorphisms
- Part II Once-punctured tori
- Part III Related topics
On hyperbolic and spherical volumes for knot and link cone-manifolds
from Part I - Hyperbolic 3-manifolds
Published online by Cambridge University Press: 10 September 2009
- Frontmatter
- Contents
- Preface
- Part I Hyperbolic 3-manifolds
- Combinatorial and geometrical aspects of hyperbolic 3-manifolds
- Harmonic deformations of hyperbolic 3-manifolds
- Cone-manifolds and the density conjecture
- Les géodésiques fermées d'une variété hyperbolique en tant que nœuds
- Ending laminations in the Masur domain
- Quasi-arcs in the limit set of a singly degenerate group with bounded geometry
- On hyperbolic and spherical volumes for knot and link cone-manifolds
- Remarks on the curve complex: classification of surface homeomorphisms
- Part II Once-punctured tori
- Part III Related topics
Summary
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.
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- Kleinian Groups and Hyperbolic 3-ManifoldsProceedings of the Warwick Workshop, September 11–14, 2001, pp. 145 - 164Publisher: Cambridge University PressPrint publication year: 2003
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