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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.05903 |
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| _version_ | 1866910906422657024 |
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| author | Arai, Katsunori |
| author_facet | Arai, Katsunori |
| contents | A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e., oriented compact surfaces with boundaries embedded in the $3$-sphere $S^{3}$. A $G$-family of racks is a set with a family of binary operations indexed by the elements of a group $G$. There are two known methods for constructing multiple group racks. One is via a $G$-family of racks. The resulting multiple group rack is called the associated multiple group rack of the $G$-family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a $G$-family of racks and a normal subgroup $N$ of $G$. We show that this construction yields multiple group racks that are neither the associated multiple group racks of any $G$-family of racks nor their abelian extensions when the right conjugation action of $G$ on $N$ is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any $G$-family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A construction of multiple group racks Arai, Katsunori Geometric Topology 57K12, 57K10 A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e., oriented compact surfaces with boundaries embedded in the $3$-sphere $S^{3}$. A $G$-family of racks is a set with a family of binary operations indexed by the elements of a group $G$. There are two known methods for constructing multiple group racks. One is via a $G$-family of racks. The resulting multiple group rack is called the associated multiple group rack of the $G$-family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a $G$-family of racks and a normal subgroup $N$ of $G$. We show that this construction yields multiple group racks that are neither the associated multiple group racks of any $G$-family of racks nor their abelian extensions when the right conjugation action of $G$ on $N$ is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any $G$-family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper. |
| title | A construction of multiple group racks |
| topic | Geometric Topology 57K12, 57K10 |
| url | https://arxiv.org/abs/2504.05903 |