A Census of Tetrahedral Hyperbolic Manifolds

Abstract

We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orientable case) tetrahedra. Our isometry classification uses certified canonical cell decompositions (based on work by Dunfield, Hoffman, Licata) and isomorphism signatures (an improvement of dehydration sequences by Burton). The tetrahedral census comes in Regina as well as SnapPy format, and we illustrate its features.

The complement of the figure-eight knot geometrically bounds

“…It is worth mentioning that a tetrahedral 3-manifold M can have many non-equivalent triangulations by regular ideal hyperbolic tetrahedra. A complete classification up to homeomorphism of orientable tetrahedral manifolds which possess a triangulation with at most 25 tetrahedra (21 tetrahedra in the non-orientable case) is given in [3], where it is also shown that all tetrahedral manifolds are arithmetic.…”

Universal Quantum Computing and Three-Manifolds

“…Table 2 establishes the list of 3-manifolds corresponding to subgroups of index d ≤ 7 of the universal group G = π 1 (S 3 \ K 0 ). The manifolds are labeled otetN n in [25] because they are oriented and built from N = 2d tetrahedra, with n an index in the table. The identification of 3-manifolds of finite index subgroups of G was first obtained by comparing the cardinality list η d (H) of the corresponding subgroup H to that of a fundamental group of a tetrahedral manifold in SnapPy table [27].…”

“…The three-dimensional space surrounding a knot K -the knot complement S 3 \ K-is an example of a three-manifold [1,24]. We will be especially interested by the trefoil knot that underlies work of the first author [15] as well as the figure-of-eight knot, the Whitehead link and the Borromean rings because they are universal (in a sense described below), hyperbolic and allow to build 3-manifolds from platonic manifolds [25]. Such manifolds carry a quantum geometry corresponding to quantum computing and (possibly informationally complete) POVMs identified in our earlier work [15,13,14].…”

Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

“…Suppose that the interior of M , denoted by Q, possesses a complete Riemannian metric with finite volume and constant sectional curvature −1. Following [14], we say that M is tetrahedral if there exists a decomposition of Q into ideal regular hyperbolic tetrahedra. Equivalently, there exists an ideal triangulation of M such that each edge class contains exactly six edges of the tetrahedra of D.…”

“…As mentioned in [1,14,18], coverings of tetrahedral manifolds yield infinite families of finite volume hyperbolic 3-manifolds whose tetrahedral complexity can be calculated exactly. More precisely the following statement holds.…”