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The volume of hyperbolic alternating link complements

Published online by Cambridge University Press:  13 January 2004

Marc Lackenby
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB. E-mail: lackenby@maths.ox.ac
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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 $v_3(t(D) - 2)/2$ and $v_3(10t(D) - 10)$, 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.

Type
Research Article
Copyright
2004 London Mathematical Society

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