On identifying hyperbolic 3-manifolds as link complements in the 3-sphere

We give a straightforward method that helps recognize when a noncompact hyperbolic 3-manifold is a link complement in the 3-sphere and automatically produces the link diagram. The method is based on converting a side-pairing to a handle decomposition

Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure

Many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. Generalizing to dimension 4, we construct a dozen examples of noncompact hyperbolic 4-manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of some of those manifolds are then shown to be complements of tori and Klein bottles in other simply-connected closed 4-manifolds. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic 4-manifolds of minimal volume. Our examples are finite covers of some of those manifolds.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-41.abs.htm