Non-equivalent hyperbolic knots
type: Item
instance of: scholarly article
title: Non-equivalent hyperbolic knots
zbMATH Open document ID: 1024.57006
DOI: 10.1016/S0166-8641(01)00240-1
author: Luisa Paoluzzi
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: The question studied in the article is the following: Given two knots \(K\) and \(K'\) in \(S^{3}\), then if the knots have homeomorphic \(n\)-fold cyclic branched covers for some values of \(n\), are the two knots \(K\) and \(K'\) equivalent? This work extends work on this question by \textit{B. Zimmermann} [Math. Ann. 311, 665-673 (1998; Zbl 0913.57008) and Topology Appl. 79, 143-157 (1997; Zbl 0888.57006)]. The main theorem is the following: Given a value \(n\geq 3\), then there exist pairs of non-equivalent hyperbolic knots with the same 2-fold and \(n\)-fold cyclic branched covers. Explicit examples of such knots are given in the article. A Conway sphere in \((S^{3},K)\) is a 2-sphere \(S^{2}\) which meets \(K\) transversally in 4 points. \(K\) is Conway reducible if \((S^{3},K)\) admits a Conway sphere such that \(S^{2}-U(K)\) is incompressible and boundary incompressible in \(S^{3}-U(K)\), where \(U(K)\) represents a tubular neighborhood of \(K\). In an article by the author [Topology Appl. 91, 245-257 (1999; Zbl 0942.57003)] it was shown that a Conway irreducible hyperbolic knot is determined by its 2-fold and \(n\)-fold cyclic branched covers for any \(n\geq 3\). So the examples here show that Conway reducible hyperbolic knots behave differently than Conway irreducible hyperbolic knots with regards to properties of their 2-fold and \(n\)-fold cyclic branched covers. A \(n\)-fold branched cyclic cover \(M\) of a knot \(K\) is ambiguous if \(M\) does not determine the knot \(K\) uniquely, in particular there exists a knot \(K'\) (non-equivalent to \(K\)) such that the \(n\)-fold branched cyclic covers of a knot \(K\) and \(K'\) are homeomorphic to \(M\). A final theorem states that a hyperbolic knot \(K\) has at most 3 ambiguous \(n\)-fold branched cyclic covers and at most two with order greater than 2.
reviewed by: Claus Ernst
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1879084
zbMATH Keywords: hyperbolic knots
zbMATH Keywords: cyclic branched covers
zbMATH Keywords: orbifolds
zbMATH Keywords: Bonahon-Siebenmann decomposition
MaRDI profile type: Publication
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full work available at URL: s0166-8641(01)00240-1
OpenAlex ID: W2047274699
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description: scientific article; zbMATH DE number 1879084