Abstract. This paper describes a general algorithm for finding the commensurator of a non-arithmetic hyperbolic manifold with cusps, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all non-arithmetic hyperbolic once-punctured torus bundles over the circle.For hyperbolic 3-manifolds, the algorithm has been implemented using Goodman's computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3-manifolds triangulated using at most 7 ideal tetrahedra, and for the complements of hyperbolic knots and links with up to 12 crossings.
Abstract. In this paper we enumerate and classify the "simplest" pairs (M, G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M .To enumerate the pairs we use a variation of Matveev's definition of complexity for 3-manifolds, and we consider only (0, 1, 2)-irreducible pairs, namely pairs (M, G) such that any 2-sphere in M intersecting G transversely in at most 2 points bounds a ball in M either disjoint from G or intersecting G in an unknotted arc. To classify the pairs our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard's program Orb and Goodman's program Snap.We determine all (0, 1, 2)-irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic meridians. For these pairs we give detailed information on hyperbolic invariants including volumes, symmetry groups and arithmetic invariants. Pictures of all hyperbolic graphs up to complexity 4 are provided. We also include a partial analysis of knots and links.The theoretical framework underlying the paper is twofold, being based on Matveev's theory of spines and on Thurston's idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of our results were obtained (or suggested) by computer investigations.
We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods 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 were already 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 non-hyperbolic 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, but they do not involve issues of numerical accuracy. MSC (2000): 57M50 (primary), 57M25 (secondary).
Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central rôle played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference with the two-dimensional case, and we will review some of the results of the combinatorial and computational approach to three-manifolds developed by different mathematicians over the last several years. MSC (2000): 57M50 (primary), 57M25 (secondary).The octahedron, denoted henceforth by O, is one of the favourite toys of every geometer, being one of the five Platonic solids. In this note we will investigate the following:Question 0.1. How many different topological spaces can be obtained by gluing together in pairs the faces of O?Question 0.1 has a rather transparent combinatorial flavour and appears to be well-suited to computer investigation, but the complete answer would be extremely difficult to obtain without the aid of some rather sophisticated geometric tools developed over the last three decades by a number of mathematicians. It is indeed mostly thanks to hyperbolic geometry that one is 1 able to show that certain gluings of O, despite being very similar to each other under many respects, are in fact distinct.There is a very natural family of lesser siblings of Question 0.1, involving (two-dimensional) polygons rather than a (three-dimensional) polyhedron (as the octahedron is), and we will show below that the answers to these lower-dimensional questions are sharply different from both a qualitative and quantitative viewpoint. More precisely, it turns out that identifying the spaces obtained by gluing together the edges of a polygon is very easy, and that the number of possible different results is very small if compared to the number of (combinatorially inequivalent) gluing patterns. On the other hand, in dimension three identifying the results is often only possible using hyperbolic geometry, and there is a rather large variety of different results. This can be viewed as a manifestation of the crucial rôle played by hyperbolic geometry in the context of three-dimensional topology, as chiefly witnessed by Thurston's geometrization.As opposed to looking at lower-dimensional analogues of Question 0.1, one can also view it as a special instance of a more general family of threedimensional problems, where one considers larger polyhedra (or finite families of polyhedra). These problems have attracted a considerable attention during the last several years, and we will include a brief survey of the main results obtained.
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