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The Volume of Hyperbolic Alternating Link Complements
Author(s) -
Lackenby Marc
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014291
Subject(s) - mathematics , invariant (physics) , hyperbolic 3 manifold , hyperbolic manifold , simplex , twist , pure mathematics , volume (thermodynamics) , relatively hyperbolic group , combinatorics , mathematical analysis , hyperbolic function , geometry , mathematical physics , physics , quantum mechanics
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 (10 t ( 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. 2000 Mathematics Subject Classification 57M25, 57N10.