Hyperbolic Graphs of Small Complexity

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Hyperbolic Graphs of Small Complexity
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 hyp...
Damian Heard, Craig Hodgson, Bruno Martelli, Carlo
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2010
Where EM
Authors Damian Heard, Craig Hodgson, Bruno Martelli, Carlo Petronio
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