Irreducible Triangulations of the Klein Bottle

✍ Scribed by Serge Lawrencenko; Seiya Negami

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 399 KB
• Volume: 70
• Category: Article
• ISSN: 0095-8956
• DOI: 10.1006/jctb.1997.9999

✦ Synopsis

We determine the complete list of the irreducible triangulations of the Klein bottle, up to equivalence, analyzing their structures.

1997 Academic Press

1. Introduction

A triangulation of a closed surface is a simple graph embedded on the surface so that each face is triangular and that any two faces have at most one edge in common. (The latter is needed only for the sphere to exclude K 3 from the spherical triangulations.) It is often regarded as a 2-simplicial complex together with its triangular faces. Two triangulations G and G$ of a closed surface F 2 are said to be equivalent if there is a homeomorphism h: F 2 Ä F 2 with h(G)=G$. In the combinatorial sense, such a homeomorphism can be thought of as an isomorphism between two graphs which induces a bijection between their faces. We shall say that two triangulations are isomorphic to each other when they are isomorphic as graphs neglecting their embeddings.

Let abc and acd be two faces which share an edge ac in a triangulation G. The contraction of ac is to delete the edge ac and to identify the path bad with bcd, shrinking the quadrilateral region bounded by the cycle abcd, as shown in Fig. 1. An edge e of G is said to be contractible if the contraction of e yields another triangulation of the surface where G is embedded.