On Vertices of Frontals in the Euclidean Plane

We investigate vertices for plane curves with singular points. As plane curves with singular points, we consider Legendre curves (respectively, Legendre immersions) in the unit tangent bundle over the Euclidean plane and frontals (respectively, fronts) in the Euclidean plane. We define a vertex using evolutes of frontals. After that we define a vertex of a frontal in the general case. It is also known that the four vertex theorem does not hold for simple closed fronts. We give conditions under which a frontal has a vertex and the four vertex theorem holds for closed frontals. We also give examples and counter examples of the four vertex theorem.