2000/12/19 by Marc Lackenby, Lackenby, Marc · 3 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DG #math.GT #msc:57M25 #msc:57N10
paper · pdf · doi:10.48550/arxiv.math/0012185
30 pages, 17 figures; contains an appendix by Ian Agol and Dylan Thurston
arxiv created 2003/06/16 · arxiv updated 2009/11/30
If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v3(t(D) - 2)/2 and v3(16t(D) - 16), where v3 is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3-manifolds, in the geometric topology. The appendix by Ian Agol and Dylan Thurston, which was written after the first version of this paper was distributed, improves the upper bound on volume to v3(10t(D) - 10). In addition, examples of alternating links are given which asymptotically achieve this bound.