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

TL;DR

This paper classifies semi-equivelar gems of PL d-manifolds embedded on surfaces with Euler characteristic -1, identifying specific types and providing constructions for each, extending previous classifications to this new surface characteristic.

Contribution

The paper extends the classification of semi-equivelar gems to surfaces with Euler characteristic -1, identifying all possible types and constructing examples for each.

Findings

Identified 12 types of semi-equivelar gems on the surface with Euler characteristic -1.

Proved that any such gem belongs to one of the specified types.

Provided explicit constructions for each type.

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 degrees 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),$ or $(4,8,16)$. Furthermore, we provide constructions that demonstrate the existence of such gems for each of the aforementioned types.