Abstract: 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 knot or link. 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 formulas for volume of the corresponding manifold modeled in the hyperbolic, spherical and Euclidean geometries.

Cite: Mednykh A.
Volumes of knots and links in spaces of constant curvature
15th International Society for Analysis, its Applications and Computation Congress, Astana, July 21-25, 2025 21-25 Jul 2025