#### Volume 21, issue 7 (2021)

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Turaev hyperbolicity of classical and virtual knots

### Colin Adams, Or Eisenberg, Jonah Greenberg, Kabir Kapoor, Zhen Liang, Kate O’Connor, Natalia Pachecho-Tallaj and Yi Wang

Algebraic & Geometric Topology 21 (2021) 3459–3482
##### 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.

##### Keywords
Turaev surface, Turaev volume, knot, hyperbolic knot, virtual knot
##### Mathematical Subject Classification
Primary: 57K10, 57K32