Turaev hyperbolicity of classical and virtual knots

Abstract

By work of W Thurston, knots and links in the $3$–sphere are known to either be torus links; or to contain an essential sphere or torus in their complement; or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic $3$–manifolds of finite volume to any classical or virtual link, even if nonhyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic $3$–manifolds associated via this Turaev construction. In the case of a classical link, we can also define the classical Turaev volume, which is the minimal volume among all the hyperbolic $3$–manifolds associated via this Turaev construction for the classical projections only. We then investigate these new invariants.