Vertex-minimal simplicial immersions of the Klein bottle in three space

Although the Klein bottle cannot be embedded inR ³, it can be immersed there, and in more than one way. Smooth examples of these immersions have been studied extensively, but little is known about their simplicial versions. The vertices of a triangulation play a crucial role in understanding immersions, so it is reasonable to ask: How few vertices are required to immerse the Klein bottle inR ³? Several examples that use only nine vertices are given in Section 3, and since any triangulation of the Klein bottle must have at least eight vertices, the question becomes: Can the Klein bottle be immersed inR ³ using only eight vertices? In this paper, we show that, in fact, eight isnot enough, nine are required. The proof consists of three parts: first exhibiting examples of 9-vertex immersions; second determining all possible 8-vertex triangulations ofK ²; and third showing that none of these can be immersed inR ³.