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

Anshu Agarwal, Biplab Basak

TL;DR

The paper addresses the classification of semi-equivelar gems of PL $d$-manifolds that admit a regular embedding on a surface with Euler characteristic $-1$, i.e., the non-orientable surface $ obreak{\#_3\RP^2}$. It reduces the problem by enumerating possible embedding types via a lemma, proves that $5$- and $4$-regular cases cannot occur for such gems, and then provides explicit constructions for twelve distinct $3$-regular embedding types, each yielding a semi-equivelar gem that represents $ obreak{\#_3\RP^2}$. The twelve types are $[(8^3);8], [(6^2,8);24], [(6^2,12);12], [(10^2,4);20], [(12^2,4);12], [(4,6,14);84], [(4,6,16);48], [(4,6,18);36], [(4,6,24);24], [(4,8,10);40], [(4,8,12);24], [(4,8,16);16]$, with explicit constructions illustrated in Figures 1–12. This extends the prior χ≥0 classifications and clarifies the landscape of semi-equivelar gems on the unique non-orientable surface of Euler characteristic $-1$, while highlighting that all resulting gems realize $ obreak{\#_3\RP^2}$ and differentiating gems from semi-equivelar maps.

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.

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

TL;DR

The paper addresses the classification of semi-equivelar gems of PL -manifolds that admit a regular embedding on a surface with Euler characteristic , i.e., the non-orientable surface . It reduces the problem by enumerating possible embedding types via a lemma, proves that - and -regular cases cannot occur for such gems, and then provides explicit constructions for twelve distinct -regular embedding types, each yielding a semi-equivelar gem that represents . The twelve types are , with explicit constructions illustrated in Figures 1–12. This extends the prior χ≥0 classifications and clarifies the landscape of semi-equivelar gems on the unique non-orientable surface of Euler characteristic , while highlighting that all resulting gems realize and differentiating gems from semi-equivelar maps.

Abstract

A semi-equivelar gem of a PL -manifold is a regular colored graph that represents the PL -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 -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 -manifolds embedded on the surface with Euler characteristic . We prove that if a semi-equivelar gem embeds regularly on the surface with Euler characteristic , then it belongs to one of the following types: or . Furthermore, we provide constructions that demonstrate the existence of such gems for each of the aforementioned types.
Paper Structure (5 sections, 7 theorems, 7 equations, 13 figures)

This paper contains 5 sections, 7 theorems, 7 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph encoded manifolds (gem)
  4. Regular embedding
  5. Main results

Key Result

Proposition 1

If $\Gamma$ is a bipartite (respectively, non-bipartite) $(d+1)$-regular colored graph representing a closed connected orientable (respectively, non-orientable) PL $d$-manifold $M$, then for each cyclic permutation $\varepsilon = (\varepsilon_0, \dots, \varepsilon_d)$ of $\Delta_d$, there exists a r

Figures (13)

  • Figure 1: Only possible semi-equivelar graph of type $(4^5)$ embedded regularly on $\#_{3} \mathbb{RP}^2$.
  • Figure 2: Embedding on $\#_3 \mathbb{RP}^2$ of gem representing $\#_{3} \mathbb{RP}^2$ of type $(8^3)$.
  • Figure 3: Embedding on $\#_3 \mathbb{RP}^2$ of gem representing $\#_{3} \mathbb{RP}^2$ of type $(6^2,8)$.
  • Figure 4: Embedding on $\#_3 \mathbb{RP}^2$ of gem representing $\#_{3} \mathbb{RP}^2$ of type $(6^2,12)$.
  • Figure 5: Embedding on $\#_3 \mathbb{RP}^2$ of gem representing $\#_{3} \mathbb{RP}^2$ of type $(10^2,4)$.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Proposition 1: ga81i
  • Proposition 2: ga81i
  • Proposition 3: ga81i
  • Proposition 4: cp90
  • Definition 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • Remark 8
  • ...and 4 more