Lifting a generic surface in 3-space to an embedded surface in 4-space

We define a `BW orientationʹ for a double point set of a generic surface in 3-space. Then we show that a given generic surface in 3-space is a projection of some embedded surface in 4-space if and only if its double point set admits a BW orientation. BW here means black/white; the BW orientation is motivated by checkerboard coloring of the complement of the surface.