Constructions of two-fold branched covering spaces

José M. Montesinos and Wilbur Carrington Whitten

Vol. 125 (1986), No. 2, 415–446

DOI: 10.2140/pjm.1986.125.415

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}.

Mathematical Subject Classification 2000

Primary: 57M12

Secondary: 57M25

Milestones

Received: 26 July 1983

Revised: 25 June 1985

Published: 1 December 1986