2019/12/19 by Colin Adams, Or Eisenberg, Jonah Greenberg +6
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics #Complement (music) #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic manifold #Hyperbolic set #Link (geometry) #Mathematical analysis #Mathematics #Physics #Pure mathematics #Relatively hyperbolic group #Torus #Volume (thermodynamics) #math.GT
paper · pdf · doi:10.2140/agt.2021.21.3459
published as Algebr. Geom. Topol. 21 (2021) 3459-3482 · 14 pages, 3 tables, 10 figures
arxiv created 2019/12/19 · openalex publication_date 2021/12/28 · arxiv updated 2022/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.