On a class of hyperbolic 3-orbifolds of small volume and small Heegaard genus associated to 2-bridge links
The authors study a certain class of \(3\)-orbifolds \(O_n(\alpha/\beta)\). The underlying space of these orbifolds is the \(3\)-sphere, and the singular set is a certain link with two specific edges added, and with specific branching orders. The link is the \(2\)-bridge link \(L(\alpha/\beta)\) (the twists are encoded in any expansion of the rational number \(\alpha/\beta\) as continued fraction), and the additional edges are the two standard (upper and lower) unknotting tunnels. The authors prove by an explicit construction that \(O_n(\alpha/\beta)\) is a quotient of the orbifold \(L_n(\alpha/\beta)\) by an action of the Klein group \(Z/2\times Z/2\), and the projection is a \(4\)-fold orbifold covering. If \(L_n(\alpha/\beta)\) is hyperbolic, the covering transformations can be chosen to be isometries, in particular \(O_n(\alpha/\beta)\) has also a hyperbolic structure and the volume of \(L_n(\alpha/\beta)\) is four times the one of \(O_n(\alpha/\beta)\). Here, \(L_n(\alpha/\beta)\) is an orbifold with underlying space the \(3\)-sphere and singular set the link \(L(\alpha/\beta)\) (with branching order \(n\)). The advantage is that for such orbifolds effective algorithms exist to decide whether they are hyperbolic and to compute their volume. The authors use this (more specifically, the program snappea), to compute the volumes for a long list of examples. They obtain some of the smallest known volumes of hyperbolic \(3\)-orbifolds.
scientific article; zbMATH DE number 1462205
scientific article