Polyhedral Embeddings and Immersions of Many Triangulated 2-Manifolds   with Few Vertices
  This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one or two handles and the Klein Bottle with one or two handles. 
Ulrich Brehm and Undine Leopold
math.MG math.GT