Hyperbolic spatial graphs in 3-manifolds

For any closed connected orientable 3-manifold M, we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S3 via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M.