Braiding and Folding Branched Covers

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