Turaev Hyperbolicity of Classical and Virtual Knots

TL;DR

This paper introduces Turaev volumes as new invariants for classical and virtual knots, associating hyperbolic 3-manifolds to links and analyzing their minimal volumes to distinguish knot types.

Contribution

It defines Turaev volume and classical Turaev volume for classical and virtual links, extending hyperbolic invariants to non-hyperbolic cases using Turaev's construction.

Findings

Turaev volume provides a new invariant for classifying links.

Classical Turaev volume can distinguish classical from virtual knots.

The invariants are computed via minimal hyperbolic volumes associated with links.

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 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 non-hyperbolic. 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.