Hyperbolic knots, links and polyhedra

Knots, links or, more generally, knotted graphs admitting a hyperbolic structure on their complement are called hyperbolic. In 1975, Robert Riley discovered the existence of a family of hyperbolic knots. Today, we know that most knots and links are hyperbolic. This statement is now a special case of the Thurston Hyperbolization Theorem. Starting with non-complete hyperbolic structure on the complement, one can produce a complete hyperbolic cone-manifold whose singular set is a given knot, link or polyhedron with prescribed cone angles. This is a way to investigate geometrical properties of the knots, links and polyhedra from the unified point of view. In this lecture, we show that cone angles and lengths of singular geodesics of cone-manifolds under consideration are related by quite simple and beautiful Sine, Cosine or Tangent rules. These identities allow to find integral formulas for the volume and Chern-Simons invariants of the manifolds in many important cases.