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The volume of hyperbolic alternating link complements
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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 v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 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 v_3(10t(D) - 10). In addition, examples of alternating links are given which asymptotically achieve this bound.
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Expansion joints in hyperbolic manifolds
Cone-deforming an ideal arc through 'expansion joints' interpolates between stacked Borromean ring complements and lantern manifolds, and yields cone deformations of highly twisted 2-bridge unknotting tunnels.
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