We enumerate all the spaces obtained by gluing in pairs the
faces of the octahedron in an orientation-reversing fashion.
Whenever such a gluing gives rise to nonmanifold points, we remove
small open neighborhoods of these points, so we actually
deal with three-dimensional manifolds with (possibly empty)
boundary.
¶ There are 298 combinatorially inequivalent gluing patterns, and
we show that they define 191 distinct manifolds, of which 132
are hyperbolic and 59 are not. All the 132 hyperbolic manifolds
have already been considered in different contexts by other authors,
and we provide here their known “names” together with
their main invariants. We also give the connected sum and JSJ
decompositions for the 59 nonhyperbolic examples.
¶ Our arguments make use of tools coming from hyperbolic geometry,
together with quantum invariants and more classical techniques
based on essential surfaces. Many (but not all) proofs
were carried out by computer.