A classification of semi-equivelar gems of PL d-manifolds on the surface with Euler characteristic -1

Abstract

A semi-equivelar gem of a PL $d$-manifold is a regular colored graph that represents the PL $d$-manifold and regularly embeds on a surface, with the property that the cyclic sequence of lengths of faces in the embedding around each vertex is identical. In \cite{bb24}, the authors classified semi-equivelar gems of PL $d$-manifolds embedded on surfaces with Euler characteristics greater than or equal to zero. In this article, we focus on classifying semi-equivelar gems of PL $d$-manifolds embedded on the surface with Euler characteristic $-1$. We prove that if a semi-equivelar gem embeds regularly on the surface with Euler characteristic $-1$, then it belongs to one of the following types: $(8^3)$, $(6^2,8)$, $(6^2,12)$, $(10^2,4)$, $(12^2,4)$, $ (4,6,14)$, $(4,6,16)$, $(4,6,18)$, $(4,6,24)$, $(4,8,10)$, $(4,8,12)$ and $(4,8,16)$. Furthermore, we provide constructions that demonstrate the existence of such gems for each of the aforementioned types.