All Links Are Sublinks of Arithmetic Links

While it is known that every closed orientable 3-manifold M contains an arithmetic link L (since the figure-eight knot complement is both arithmetic and universal), Theorem 2 asserts that L can be chosen so that M \ L is homeomorphic to the complement of a link in S3. Recall that a link L in S3 (resp. in M) is hyperbolic if S3 \L (resp. M \L) is homeomorphic to H3/Γ, where H3 is hyperbolic 3-space and Γ a discrete, torsion-free, finite covolume subgroup of PSL2(C). We say that L is arithmetic if Γ can be chosen commensurable with a Bianchi group PSL2(Om), where Om is the integers of the imaginary quadratic number field Q( √−m) (see [R] for a more general discussion). Finally, L is a sublink of J if it is a union of components of J .