Constructions of two-fold branched covering spaces
By equivariantly pasting together exteriors of links in \(S^ 3\) that are invariant under several different involutions of \(S^ 3\), the authors construct closed orientable 3-manifolds that are two-fold branched covering spaces of \(S^ 3\) 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 \(M/h\cong S^ 3\). For example, a trefoil knot is strongly invertible and periodic, and one can glue its exterior to the exterior of a trivial knot in various ways to construct manifolds that are branched covering spaces of \(S^ 3\) in two different ways; these new manifolds are obviously contructed by surgery on a trefoil knot (\(1/n\)-surgery of \(2/n\)-surgery (\(n\) odd)). Using the theory of characteristic submanifolds for Haken manifolds with incompressible boundary components and the work of \textit{R. Hartley} [Math. Z. 171, 175-185 (1980; Zbl 0411.57002)], the authors prove that doubles, \(D(K,\rho)\), of prime knots that are not strongly invertible are characterized by their two-fold branched covering spaces, when \(\rho\) \(\neq 0\). If, however, \(K\) is strongly invertible, then the manifold branch covers distinct knots. Also, the double cover of \(D(K,\rho)\) characterizes \(D(K,\rho)\) among doubled knots, when \(K\) is prime and \(\rho\neq 0\). Finally, the authors characterize the type of a prime knot by the double covers of the doubled knots, \(D(K;\rho,\eta)\) and \(D(K^*;\rho,\eta)\), of \(K\) and its mirror image \(K^*\), when \(\rho\) and \(\eta\) are fixed, with \(\rho\neq 0\) and \(\eta\in \{-1,1\}\).
two-fold branched covering spaces of \(S^ 3\)
scientific article; zbMATH DE number 3895953