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Combinatorial and geometric methods in topology. (English) Zbl 1179.57027
This readable article discusses the enumeration of 3-manifolds constructed by pairwise identification of the faces of an octahedron, and by pairwise identification of the faces of a truncated octahedron. The manifolds constructed in this way are enumerated according to their boundaries and their hyperbolicity. The article also contains some discussion of the enumeration of surfaces obtained by identifying in pairs the edges of a polygon, and 3-manifolds obtained by gluing together in pairs the faces of a fixed number of tetrahedra.
It is worthwhile pointing out that the number of combinatorially inequivalent patterns for $$P_k$$ (discussed in Section 1 of the paper) is equal to the number of distinct degree $$k$$ chord diagrams considered up to reflection, and is therefore known [see A. Stoimenow, Discrete Math. 218, No. 1–3, 209–233 (2000; Zbl 0953.57006)].
##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57M25 Knots and links in the $$3$$-sphere (MSC2010)
##### Keywords:
enumeration of 3-manifolds; hyperbolic geometry; octahedron
SnapPea; Orb
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