Sphere eversion from the viewpoint of generic homotopy

In 1958, Smale proved that any immersions of S2 to R3 are regularly homotopic. This means that we can turn an embedded sphere in R3 inside out by a regular homotopy. After Smale showed his result without visualization, many people visualized sphere eversions, in various ways. In this paper, we construct a sphere eversion by lifting a “simple” generic homotopy of S2 to R2 to a generic regular homotopy of S2 to R3. By doing so, our eversion is simple in terms of deformation of the contour generators of immersed spheres. We also visualize the 3-dimensional interlinking of the contour generators and the self-intersections of each stage of immersed spheres.