|
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number. The hyperbolic volume of a non-hyperbolic knot is often defined to be zero. By Mostow rigidity, the volume is a topological invariant of the link;〔 as a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.〔 There are only finitely many hyperbolic knots with the same volume.〔 A mutation of a hyperbolic knot will have the same volume,〔.〕 so it is possible to concoct examples with the same volume; indeed, there are arbitrarily large finite sets of distinct knots with equal volumes.〔.〕 In practice, hyperbolic volume has proven very effective in distinguishing knots, utilized in some of the extensive efforts at knot tabulation. Jeffrey Weeks's computer program SnapPea is the ubiquitous tool used to compute hyperbolic volume of a link.〔.〕 More generally, the hyperbolic volume may be defined for any hyperbolic 3-manifold. The Weeks manifold has the smallest possible volume of any closed manifold (a manifold that, unlike link complements, has no cusps); its volume is approximately 0.9427.〔.〕 ==List== * Figure-eight knot = 2.0298832 * Three-twist knot = 2.82812 * Stevedore knot (mathematics) = 3.16396 * 6₂ knot = 4.40083 * Endless knot = 5.13794 * Perko pair = 5.63877 * 6₃ knot = 5.69302 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「hyperbolic volume」の詳細全文を読む スポンサード リンク
|