On Branched Coverings of S3

Abstract

In [3] Fox studied a certain class of irregular coverings of S3 branched along some knot or link which turned out to be homotopy spheres. By a simple geometric construction, it is shown in this paper that these homotopy spheres are just 3-spheres, provided that the group of the knot or link k in question cannot be generated by a number of Wirtinger generators smaller than the minimal number of bridges of this knot or link. The knots and links with two bridges provide examples for such coverings. In the covering sphere there is a link covering k. With the help of braid automorphisms, can be determined. Figure 3 shows a link in a 5-sheeted covering over k = 41. Links over 31 and 61 in 3-sheeted coverings were determined by Kinoshita [5] by a different method. The method used here is applicable to these cases and confirms his results.