REPRESENTATIVITY AND NON-CONTRACTIBLE SEPARATING CYCLES OF EMBEDDINGS ON THE TRIPLE TORUS

Abstract

We consider Ellingman’s and Zhao’s method of proving that every 4 representative graph embedding on the double torus contains a Non-Contractible Separating Cycle (NSC). They proved this main result by considering critical embeddings; which are embeddings that are very close to having NSCs. We adopt the method in proving an extension of the same theorem to a surface of one genus higher; the triple torus. The method works efficiently in proving our main result that every 4 representative embedding on the triple torus contains two NSCs which separates the triple torus into 3 connected components, namely punctured tori, two of them with one boundary circle and one with two boundary circles. Our results are obtained by employing equivalence of embeddings and homeomorphism of surfaces to Ellingman and Zhao’s method.