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The 191 orientable octahedral manifolds. (English) Zbl 1171.57017

In the paper under review, with the aid of computers, the authors enumerate all compact orientable 3-manifolds obtained from the octahedron by gluing pairs of the faces. If such a gluing gives rise to non-manifold points, they remove small open neighborhoods to get a compact orientable manifold with (possibly empty) boundary. Actually they show that 191 distinct manifolds appear in such a way, 17 of which are closed, 30 have single torus boundary, 7 have two tori as boundary, 79 have genus two boundary, and the others, 56, have genus 3 surface as boundary. Among them, all the 132 hyperbolic manifolds are identified with their own “names” already given by other authors. Also presented are the connected sum and JSJ decompositions for the 59 non-hyperbolic examples. Most of the enumerating process was carried out by a computer, mainly based on two computer programs; “Orb” for finding hyperbolic structures on hyperbolic 3-orbifolds and 3-manifolds developed by the first author, and “Three-manifold Recognizer” for recognition of 3-manifolds developed by S.V. Matveev and V.V. Tarkaev. The authors found certain problematic (i.e., not distinguished by the programs) pairs; these were distinguished by theoretical means by hand.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes

Software:

SnapPea; Orb; Snap
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