Vertex-minimal simplicial immersions of the Klein bottle in three space (Q1332620)

The author shows 9 to be the minimal number of vertices of a polyhedral immersion of the Klein bottle into \(\mathbb{R}^ 3\). First he exhibits some interesting 9-vertex immersions which have been constructed with the aid of a graphics workstation. In a next step he finds the 6 combinatorial types of 8-vertex triangulations of the Klein bottle. Finally he shows that none of them may be immersed polyhedrally into \(\mathbb{R}^ 3\) by investigating the homotopy type of the double sets of possible immersions. All these investigations are done without using a computer.