Hyperbolic graphs of small complexity. (English) Zbl 1207.57024

In the paper under review, the authors consider trivalent graphs embedded in closed orientable 3-manifolds, and give a complete list of such pairs of complexity at most 5 satisfying certain irreducibility conditions. (In the case of complexity 5, graphs having components without vertices are forbidden.) The complexity they use is a variation of S. V. Matveev’s well-known complexity for 3-manifolds, originally defined in [Acta Appl. Math. 19, No. 2, 101–130 (1990; Zbl 0724.57012)]. To obtain the list, they first generate all efficient triangulations with at most 5 tetrahedra satisfying some minimality criteria. From the triangulations so-obtained, (trivalent graph, 3-manifold)-pairs are constructed. Then the authors use some computer programs, called Orb, SnapPea, and Snap, to identify and classify the hyperbolic ones among the pairs. Detailed hyperbolic invariants and properties are calculated for them. Also, for the remaining non-hyperbolic pairs, the authors give a classification and a complete description by using classical topological techniques.


57M50 General geometric structures on low-dimensional manifolds
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
05C30 Enumeration in graph theory
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57M15 Relations of low-dimensional topology with graph theory


Zbl 0724.57012


Haskell; SnapPea; Geo; Snap; Orb
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