6-vertex theorem for closed planar curve which bounds an immersed surface with non-zero genus
Masaaki Umehara · Nagoya Mathematical Journal · 1994
A vertex of a planar curve γ of class is a point which attains a local maximum or minimum of its curvature function. By the definition, the number of vertices are even whenever it is finite. As a generalization of famous four vertex theorem, Pinkall [P] showed that a closed curve γ has at least 4 vertices if it bounds an immersed surface, and he conjectured that γ has at least 4g + 2 vertices when the surface has genus g.