Birth of bitangencies in a family of surfaces in R4

In this paper we study transitions in the number of bitangencies in a one-parameter family of closed surfaces immersed in R4. We begin by calculating the linking number between the parabolic curve of the surface and a particular normal vector field along the curve, known as the flecnodal normals. We find that there is a global formula relating the linking number to the number of bitangencies, the number of double points, and the normal Euler number of the surface. Then we calculate all possible transitions in the number of bitangencies. The transitions split into two types, local and multi-local, which we look at separately. The transitions occur at codimension 1 points, specifically full flecnodal points, degenerate bitangencies and degenerate double points. In all cases, we calculate how changes in the set of bitangencies correspond to changes in the geometry of the surface.