On \(\pi\)-hyperbolic knots with the same 2-fold branched coverings
This paper studies to what extent a \(\pi\)-hyperbolic knot is determined by its \(2\)-fold branched covering. A knot \(K\) is called \(\pi\)-hyperbolic if there is a singular hyperbolic structure on the \(3\)-sphere with singular locus \(K\) and cone angle \(\pi\) at each singular point. A knot is known to be \(\pi\)-hyperbolic iff its \(2\)-fold branched covering is hyperbolic and the covering involution is an isometry. The main result of the paper states that there are at most nine conjugacy classes of nonfree involutions in the group of orientation-preserving isometries of a closed oriented hyperbolic \(\mathbb Z_2\)-homology \(3\)-sphere. In particular, there are at most nine \(\pi\)-hyperbolic knots with the same \(2\)-fold branched covering. According to the author, the largest known number of \(\pi\)-hyperbolic knots with the same \(2\)-fold branched covering is four. The proof depends on a characterization of finite Sylow \(2\)-subgroups of the orientation-preserving diffeomorphism group of a \(\mathbb Z_2\)-homology \(3\)-sphere.
scientific article; zbMATH DE number 1463209