A classification of semi-equivelar gems on the double torus
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918248977530880 |
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| author | Agarwal, Anshu Basak, Biplab Ghosh, Debolina |
| author_facet | Agarwal, Anshu Basak, Biplab Ghosh, Debolina |
| contents | A \emph{semi-equivelar gem} of a PL $d$-manifold is a regular colored graph that represents the manifold and admits a regular embedding on a surface, such that the cyclic sequence of face degrees around each vertex is identical. In [1,4], semi-equivelar gems of PL $d$-manifolds embedded on surfaces with Euler characteristic $χ\geq -1$ were classified.
In this paper, we extend this classification to semi-equivelar gems embedded on the double torus. We show that any such gem must belong to one of the following 31 types: $(4^5)$, $(6^4)$, $(4^3,6)$, $(4^3,8)$, $(4^3,12)$, $(4^2,6^2)$, $(4,6,4,6)$, $(4^2,8^2)$, $(4,8,4,8)$, $(8^3)$, $(10^3)$, $(6^2,8)$, $(6^2,10)$, $(6^2,12)$, $(6^2,18)$, $(10^2,4)$, $(12^2,4)$, $(16^2,4)$, $(8^2,6)$, $(12^2,6)$, $(4,6,14)$, $(4,6,16)$, $(4,6,18)$, $(4,6,20)$, $(4,6,24)$, $(4,6,36)$, $(4,8,10)$, $(4,8,12)$, $(4,8,16)$, $(4,8,24)$, and $(4,10,20)$. Furthermore, we provide explicit constructions of semi-equivelar gems realizing each of these types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A classification of semi-equivelar gems on the double torus Agarwal, Anshu Basak, Biplab Ghosh, Debolina Geometric Topology 57Q15, Secondary 05C15, 05C10, 52B70, 52C20 A \emph{semi-equivelar gem} of a PL $d$-manifold is a regular colored graph that represents the manifold and admits a regular embedding on a surface, such that the cyclic sequence of face degrees around each vertex is identical. In [1,4], semi-equivelar gems of PL $d$-manifolds embedded on surfaces with Euler characteristic $χ\geq -1$ were classified. In this paper, we extend this classification to semi-equivelar gems embedded on the double torus. We show that any such gem must belong to one of the following 31 types: $(4^5)$, $(6^4)$, $(4^3,6)$, $(4^3,8)$, $(4^3,12)$, $(4^2,6^2)$, $(4,6,4,6)$, $(4^2,8^2)$, $(4,8,4,8)$, $(8^3)$, $(10^3)$, $(6^2,8)$, $(6^2,10)$, $(6^2,12)$, $(6^2,18)$, $(10^2,4)$, $(12^2,4)$, $(16^2,4)$, $(8^2,6)$, $(12^2,6)$, $(4,6,14)$, $(4,6,16)$, $(4,6,18)$, $(4,6,20)$, $(4,6,24)$, $(4,6,36)$, $(4,8,10)$, $(4,8,12)$, $(4,8,16)$, $(4,8,24)$, and $(4,10,20)$. Furthermore, we provide explicit constructions of semi-equivelar gems realizing each of these types. |
| title | A classification of semi-equivelar gems on the double torus |
| topic | Geometric Topology 57Q15, Secondary 05C15, 05C10, 52B70, 52C20 |
| url | https://arxiv.org/abs/2512.13135 |