On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links |
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Authors: | Mattia Mecchia and Bruno Zimmermann |
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Institution: | (1) Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, 34100 Trieste, Italy |
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Abstract: | We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two
standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion
of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the
volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that
an orbifoldO(α/β) has a uniqueD
2-covering by an orbifoldℒ
n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only ifℒ
n(α/β) is hyperbolic. As the volumes of the orbifoldsℒ
n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal
with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels). |
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Keywords: | 1991 Mathematics Subject Classification" target="_blank">1991 Mathematics Subject Classification 57M12 57M25 57M50 |
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