Vertex-Minimal Simplicial Immersions of the Klein Bottle in Three Space Davide P. Cervone Abstract Although the Klein bottle can not be embedded in R 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 in R 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 in R 3 using only eight vertices? In this paper, we show that, in fact, eight is not enough, nine are required. The proof consists of three parts: first exhibiting examples of 9-vertex immersions; second determining all possible 8-vertex triangulations of K 2 ; and third showing that none of these can be immersed in R 3 . 1. Introduction Smooth embeddings and immersions of surfaces have been studied extensively, but their simplicial counterparts are not so well understood. Two important questions concerning simplicial surfaces are: what is the minimum number of vertices required to triangulate the surface; and what is the minimum number of vertices needed to produce an embedding or an immersion of the surface into Euclidean space? For the sphere, the answer to both questions is four: a tetrahedron is a triangulation of the sphere using only four vertices, and it can be embedded in R 3 . For a torus, the minimum needed for a triangulation can be shown to be seven (see section 4), and a somewhat surprizing fact is that this triangulation also can be embedded in R 3 using straight edges and planar faces [9]. For a M¨obius band, five vertices are needed for a