Abstract
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.
Similar content being viewed by others
References
Banchoff, T. F.: Triple points and singularities of projections of smoothly immersed surfaces,Proc. Amer. Math. Soc. 46 (1974), 402–406.
Banchoff, T. F.: Triple points and surgery of immersed surfaces,Proc. Amer. Math. Soc. 46 (1974), 407–413.
Banchoff, T. F. and Takens, Floris: Height functions with three critical points,Ill. J. Math. 76 (1975), 325–335.
Bokowski, J. and Brehm, U.: A polyhedron of genus 4 with minimal number of vertices and maximal symmetry,Geom. Dedicata 29 (1989), 53–64.
Brehm, U.: Polyeder mit zehn Ecken vom Geschlecht drei,Geom. Dedicata 11 (1981), 119–124.
Brehm, U.: A maximally symmetric polyhedron of genus 3 with 10 vertices,Mathematika 34 (1987), 237–242.
Brehm, U.: How to build minimal polyhedral models of the Boy surface,Math. Intelligencier 12 (1990), 51–56.
Coxeter, H. S. M.: Map-coloring problems,Scripta Math. 23 (1957), 11–25.
Császár, A.: A polyhedron without diagonals,Acta. Sci. Math. 13 (1949), 140–142.
Franklin, P.: A six color problem,J. Math. Sci. 13 (1934), 363–369.
Möbius, A. F.:Gesammelte Werke, vol. II, Leipzig (1886), 482, 521, 553.
Pinkall, U.: Regular homotopy classes of immersed surfaces,Topology 24 (1985), 421–434.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cervone, D.P. Vertex-minimal simplicial immersions of the Klein bottle in three space. Geom Dedicata 50, 117–141 (1994). https://doi.org/10.1007/BF01265307
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01265307