Geometry of knots and links

Abstract

We give an overview of recent results on the geometry of knots and links. More precisely, we investigate the existence of hyperbolic, spherical or Euclidean structure on various cone manifolds whose underlying space is the three-dimensional sphere and whose 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 volumes of the corresponding cone manifolds.