On hyperbolic knots with the same m-fold and n-fold cyclic branched coverings

Abstract

We have proved in previous work that, for any pair of different integers m > n > 2 (respectively m > n ⩾ 2) which are not coprime, a hyperbolic (respectively 2π n-hyperbolic ) knot is determined by its m-fold and n-fold cyclic branched coverings; also, if n is not a power of two, there exist at most two hyperbolic or 2π n-hyperbolic knots with the same n-fold cyclic branched covering. In the present paper, for any pair of coprime integers m, n > 2, we construct the first examples of different hyperbolic knots having the same m-fold and also the same n-fold cyclic branched coverings; in fact there exist infinitely many different pairs of such knots. We construct also infinitely many triples of different π-hyperbolic knots such that the three knots of each triple have the same 2-fold branched covering; these coverings form an infinite series of hyperbolic homology 3-spheres starting from the spherical Poincaré homology 3-sphere. The question remains open how many different π-hyperbolic knots can have the same 2-fold branched covering (there are arbitrarily many hyperbolic knots with this property).