The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.