Vol. 125, No. 2, 1986

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Constructions of two-fold branched covering spaces

José M. Montesinos and Wilbur Carrington Whitten

Vol. 125 (1986), No. 2, 415–446

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∕hS3. 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 ρ0. 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 ρ0 and η ∈{−2,2}.

Mathematical Subject Classification 2000
Primary: 57M12
Secondary: 57M25
Received: 26 July 1983
Revised: 25 June 1985
Published: 1 December 1986
José M. Montesinos
Wilbur Carrington Whitten
1620 Cottontown Road
Forest VA 24551
United States