Defining and distinguishing "Twisted" lens spaces

The lens space L_n,k may be constructed by identifying certain pairs of faces of the suspension of a regular n-gon, so that the cone points are glued together. If the faces are paired together as in the construction of L_n,k but the faces are identified with a twist, the resulting complex will be a three-manifold for certain k. Many of these manifolds are shown to be of distinct homotopy type, by an analysis of the number of distinct, irreducible representations of the fundamental group into SO(3).

These manifolds are shown to be Seifert fibered, and are completely distinguished by an analysis of the base orbifold structure. Further, they are shown to be homeomorphic to the cyclic covers of S³, branched over the trefoil knot.

This geometric procedure generalizes to construct other classes of manifolds, including certain Brieskorn manifolds— namely, the cyclic coverings of S³, branched over an arbitrary torus knot. By similar methods, many of these manifolds are also shown to be distinct. In particular, it is shown that the cyclic covers of S³, branched over a fixed torus knot, are distinct.