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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.00830 |
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| _version_ | 1866915528388378624 |
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| author | Clément, Adrien |
| author_facet | Clément, Adrien |
| contents | Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology group. These groups are useful in applications, but hard to compute. In this paper, we focus on Alexander quandles over a cyclic group $\mathbb{Z}_n$. By using explicit rewriting techniques, we show that the structure group of such a quandle injects into $\mathbb{Z}^m \ltimes \mathbb{Z}_n$ if $m$ is its number of orbits. This allows us to compute its second quandle homology group, and find that the torsion part depends only on $m$ and $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure groups and second homology groups of linear Alexander quandles Clément, Adrien Group Theory 57K12 (Primary) 20F05, 16S15, 16T25 (Secondary) Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology group. These groups are useful in applications, but hard to compute. In this paper, we focus on Alexander quandles over a cyclic group $\mathbb{Z}_n$. By using explicit rewriting techniques, we show that the structure group of such a quandle injects into $\mathbb{Z}^m \ltimes \mathbb{Z}_n$ if $m$ is its number of orbits. This allows us to compute its second quandle homology group, and find that the torsion part depends only on $m$ and $n$. |
| title | Structure groups and second homology groups of linear Alexander quandles |
| topic | Group Theory 57K12 (Primary) 20F05, 16S15, 16T25 (Secondary) |
| url | https://arxiv.org/abs/2510.00830 |