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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

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.

Turaev surface, Turaev volume, knot, hyperbolic knot, virtual knot
Mathematical Subject Classification
Primary: 57K10, 57K32
Received: 31 March 2020
Revised: 26 October 2020
Accepted: 9 November 2020
Published: 28 December 2021
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
Cambridge, MA
United States
Yi Wang
Department of Mathematics
University of Pennsylvania
Philadelphia, PA
United States