#### 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.

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##### Keywords
Turaev surface, Turaev volume, knot, hyperbolic knot, virtual knot
##### Mathematical Subject Classification
Primary: 57K10, 57K32
##### Publication
Revised: 26 October 2020
Accepted: 9 November 2020
Published: 28 December 2021
##### Authors
 Colin Adams Department of Mathematics Williams College Williamstown, MA United States Or Eisenberg Boulder, CO United States Jonah Greenberg New York, NY United States Kabir Kapoor Department of Mathematics University of California, Berkeley Berkeley, CA United States Zhen Liang Department of Mathematics Boston College Chestnut Hill, MA United States Kate O’Connor Department of Mathematics Rice University Houston, TX United States Natalia Pachecho-Tallaj Department of Mathematics MIT Cambridge, MA United States Yi Wang Department of Mathematics University of Pennsylvania Philadelphia, PA United States