Non-existence of polyhedral immersions of triangulated surfaces in ${\mathbb{R}}^3$

Abstract

We present and apply a method for disproving the existence of polyhedral immersions in ${\mathbb{R}}^3$ of certain triangulations on non-orientable surfaces. In particular, it is proved that neither of the two vertex-minimal, neighborly 9-vertex triangulations of the non-orientable surface of genus 5 are realizable as immersed polyhedral surfaces in ${\mathbb{R}}^3$.

25 pages, 11 figures