Folding Branched Covers of the $3$-Sphere Branched over Knots

Abstract

A folding of a branched cover of the 3-sphere that is branched over a knot is a continuous map of the cover into the product of the sphere with a disk that has the property that the projection onto the sphere factor induces the covering. Moreover, away from the branch set, the map is a general position immersion. Cyclic branched covers can be folded so that the map is an embedding when the disk factor is 2-dimensional. Dihedral branched covers can also be folded. In as much as possible, the foldings that are presented are quite detailed. In particular, the paper focuses upon a folding of the dihedral cover of the 3-sphere that is branched along a torus knot of type (2,5). The cover also is homeomorphic to the $3$-sphere.