On hyperbolic knots with homeomorphic cyclic branched coverings (Q1271212)

This interesting paper is concerned with the problem of determining knots by their cyclic branched coverings. More precisely, the main questions are the following: How many different knots can have the same \(n\)-fold cyclic branched covering? For which values of \(n\) is a knot determined by its \(n\)-fold cyclic branched covering? How many cyclic branched coverings are sufficient to determine a knot? The author proves beautiful results which answer the questions above for the class of \(2\pi/n\)-hyperbolic (and whence hyperbolic for \(n\geq 4)\) knots. For an integer \(n\geq 2\), a knot or link \(K\) is called \(2\pi/n\)-hyperbolic if the 3-sphere containing it admits a hyperbolic metric which becomes singular, with an angle of \(2\pi/n\), around the components of \(K\). The main theorems of the paper are the following: Suppose that \(n\) is not a power of 2. Then two inequivalent \(2\pi/n\)-hyperbolic knots \(K\) and \(F\) have the same \(n\)-fold cyclic branched covering \(M\) if and only if \((S^3,K)\) and \((S^3,F)\) are the \(n\)-fold cyclic branched coverings along the unknotted components \(F'\) (with \(K\) as the preimage of \(K')\) and \(K'\) (with \(F\) as the preimage of \(F')\) of a \(2\pi/n\)-hyperbolic 2-component link \(L=K'\cup F'\) in \(S^3\), which is not symmetric in \(K'\) and \(F'\) (i.e. there is no homeomorphism of \(S^3\) interchanging \(K'\) and \(F')\). Moreover, there exist at most two inequivalent \(2\pi/n\)-hyperbolic knots with the same \(n\)-fold cyclic branched covering. Finally, suppose that \(m>n\geq 2\) are not coprime. Then any \(2\pi/n\)-hyperbolic knot \(K\) is determined by its \(m\)-fold and \(n\)-fold cyclic branched coverings.