欧拉示性数为-1的曲面上PL $d$-流形的半等角宝石的分类
A classification of semi-equivelar gems of PL $d$-manifolds on the surface with Euler characteristic $-1$
AI总结：
本文对欧拉示性数为-1的曲面上PL d-流形的半等角宝石进行分类，证明了其类型限于12种特定度数序列，并给出存在性构造。
摘要：
PL $d$-流形的半等角宝石是一种正则着色图，它表示该PL $d$-流形并正则嵌入到曲面上，且具有这样的性质：嵌入中每个顶点周围面的度数的循环序列是相同的。在文献\cite{bb24}中，作者对嵌入到欧拉示性数大于或等于零的曲面上的PL $d$-流形的半等角宝石进行了分类。在本文中，我们专注于对嵌入到欧拉示性数为$-1$的曲面上的PL $d$-流形的半等角宝石进行分类。我们证明，如果一个半等角宝石正则嵌入到欧拉示性数为$-1$的曲面上，那么它属于以下类型之一：$(8^3), (6^2,8), (6^2,12), (10^2,4), (12^2,4),$ $ (4,6,14), (4,6,16), (4,6,18), (4,6,24), (4,8,10), (4,8,12),$ 或 $(4,8,16)$。此外，我们提供了构造，证明了上述每种类型的此类宝石的存在性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
