The volume of hyperbolic alternating link complements

Abstract

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(10t(D) − 10), 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. 2000 Mathematics Subject Classification 57M25, 57N10.