We investigate the existence of hyperbolic, spherical or Euclidean structure on cone-manifolds whose underlying space is the three-dimensional sphere and singular set is a given two-bridge knot. For two-bridge knots with not more than 7 crossings we present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone-manifolds. Then these identities are used to produce exact integral formulae for the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.

中文翻译：

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恒定曲率空间中两桥锥流形的体积

我们研究了锥流形上双曲，球形或欧几里得结构的存在，其基础空间是三维球体，奇异集是给定的两桥结。对于不超过7个交叉点的两桥结，我们给出了三角恒等式，其中包括奇异测地线的长度和此类锥形流形的锥角。然后，这些恒等式可用于为在双曲线，球面和欧几里得几何模型中建模的相应锥流形的体积生成精确的积分公式。