Constructions of two-fold branched covering spaces

Abstract

By equivariantly pasting together exteriors of links in S3 that are invariant under several different involutions of S3, we construct closed orientable 3-manifolds that are two-fold branched covering spaces of S3 in distinct ways, that is, with different branch sets. Sufficient conditions are given to guarantee when the constructed manifold M admits an induced involution, h, and when M∕h≅S3. Using the theory of characteristic submanifolds for Haken manifolds with incompressible boundary components, we also prove that doubles, D(K,ρ), of prime knots that are not strongly invertible are characterized by their two-fold branched covering spaces, when ρ≠0. If, however, K is strongly invertible, then the manifold branch covers distinct knots. Finally, the authors characterize the type of a prime knot by the double covers of the doubled knots, D(K;ρ,η) and D(K∗;ρ,η), of K and its mirror image K∗ when ρ and η are fixed, with ρ≠0 and η ∈{−2,2}.