Non-equivalent hyperbolic knots
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.
scientific article; zbMATH DE number 1879084
scientific article