Complement of a Torus in the 3-Sphere

Here is a representation of a continuous transformation from the complement of a torus in a 3-sphere, to two intertwined annulus. This is an example to help understand Alexander duality.

The 3-sphere is represented with the Gott projection. In this projection, the 3-sphere is divided in two hemispheres (north and south), which are virtually glued together on their boundary. This projection can also be used for the 2-sphere, as shown below with the Gott projection of the earth.

Here is a torus in a 3-sphere (left), its complement (middle) and a cut-view of its complement (right).

Here are some intermediate steps to transform the complement into two annulus. The idea is to make room for the inside annulus, in the right hemisphere, and then push the room walls outside the right hemisphere (so into the left one).