On $\pi$ -hyperbolic knots with the same 2-fold branched coverings

Abstract

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in \(S^3\) and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation \(K_1\approx K_2\) if \(M(K_1)\) is homeomorphic to \(M(K_2)\). It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related.