E. A. Fominykh

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, and 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.

4-colored Graphs and Knot/Link Complements

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.

ON COMPLEXITY OF THREE-DIMENSIONAL HYPERBOLIC MANIFOLDS WITH GEODESIC BOUNDARY

The nonintersecting classes ℋp,q are defined, with p, q ∈ ℕ and p ≥ q ≥ 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ∈ ℋp,q, then the complexity c(M) and the Euler characteristic χ(M) of M are related by the formula c(M) = p−χ(M). The classes ℋq,q, q ≥ 1, and ℋ2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ℋ3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ɛ-invariants of manifolds.

Dehn surgeries on the figure eight knot: an upper bound for complexity

We establish an upper bound ω(p/q) on the complexity of the manifolds obtained by p/qsurgeries on the figure eight knot. It turns out that in case ω(p/q) ⩽ 12 the bound is sharp.

Two-sided bounds for the complexity of hyperbolic three-manifolds with geodesic boundary

We construct an infinite family of hyperbolic three-manifolds with geodesic boundary that generalize the Thurston and Paoluzzi-Zimmermann manifolds. For the manifolds of this family, we present two-sided bounds for their complexity.

Three-Dimensional Manifolds with Poor Spines

A special spine of a 3-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact 3-manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the manifold M have the same number of true vertices. We prove that the complexity of a compact hyperbolic 3-manifold with totally geodesic boundary that has a poor special spine with two 2-components and n true vertices is equal to n. Such manifolds are constructed for infinitely many values of n.

Cusped hyperbolic 3-manifolds of complexity 10 having maximum volume

We give a complete census of orientable cusped hyperbolic 3-manifolds obtained by gluing at most ten regular ideal hyperbolic tetrahedra. Although the census is exhaustive, the question of nonhomeomorphism remains open for some pairs of manifolds with one, two, and three cusps.

A complete description of normal surfaces for infinite series of 3-manifolds.

The set of all normal surfaces in a 3-manifold is a partial monoid under addition with a minimal generating set of fundamental surfaces. The available algorithm for finding the system of fundamental surfaces is of a theoretical nature and admits no implementation in practice. In this article, we give a complete and geometrically simple description for the structure of partial monoids for normal surfaces in lens spaces, generalized quaternion spaces, and Stallings manifolds with fiber a punctured torus and a hyperbolic monodromy map.

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

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequence, we compute the exact value of this invariant for an infinite family of such manifolds.

Complexity of virtual 3-manifolds

Virtual