Constructions of two-fold branched covering spaces
type: Item
instance of: scholarly article
title: Constructions of two-fold branched covering spaces
zbMATH Open document ID: 0562.57004
author: Wilbur Whitten
author: José María Montesinos-Amilibia
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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 \(\eta\in \{-1,1\}\).
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 3895953
zbMATH Keywords: strongly invertible knot
zbMATH Keywords: symmetric links
zbMATH Keywords: exteriors of links
zbMATH Keywords: involutions of \(S^ 3\)
zbMATH Keywords: two-fold branched covering spaces of \(S^ 3\)
zbMATH Keywords: surgery on a trefoil knot
zbMATH Keywords: characteristic submanifolds
zbMATH Keywords: Haken manifolds
zbMATH Keywords: prime knots
zbMATH Keywords: doubled knots
MaRDI profile type: Publication
full work available at URL: pjm.1986.125.415
OpenAlex ID: W2041551695
DOI: 10.2140/PJM.1986.125.415
description: scientific article; zbMATH DE number 3895953