Hyperbolic graphs of small complexity

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 two 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 comp..