A combinatorial genesis of the right-angled relations in Artin's classical braid groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911565865811968 |
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| author | Alvarado-Garduño, Omar González, Jesús Kahle, Matthew |
| author_facet | Alvarado-Garduño, Omar González, Jesús Kahle, Matthew |
| contents | The configuration space $\text{UC}(n,p\times q)$ of $n$ unlabelled non-overlapping unit squares in a $p\times q$ rectangle is known to recover the homotopy type of the classical configuration space of $n$ unlabelled points in the plane, provided $\min\{p,q\}\geq n$. Thus the fundamental group $B_n(p\times q)$ of $\text{UC}(n,p\times q)$ yields a $(p,q)$-approximation of Artin's classical braid group $B_n$. We describe a right-angled Artin group presentation for $B_n(p\times q)$ in cases where $\text{UC}(n,p\times q)$ is known to be aspherical. When $\min\{p,q\}=2$, our presentation agrees with Artin's classical presentation for $B_n$ removing the Artin-Tits relations. This allows us to deduce the value of the Lusternik-Schnirelmann category of the corresponding aspherical spaces $\text{UC}(n,p\times q)$, as well as the values of all their $k$-sequential topological complexities, both in the classical (Rudyak et al.) and distributional (Dransihnikov et al.) contexts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_12201 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A combinatorial genesis of the right-angled relations in Artin's classical braid groups Alvarado-Garduño, Omar González, Jesús Kahle, Matthew Algebraic Topology 20F36, 55R80, 57M05, 57Q70 (20F65, 57M15, 82B26) The configuration space $\text{UC}(n,p\times q)$ of $n$ unlabelled non-overlapping unit squares in a $p\times q$ rectangle is known to recover the homotopy type of the classical configuration space of $n$ unlabelled points in the plane, provided $\min\{p,q\}\geq n$. Thus the fundamental group $B_n(p\times q)$ of $\text{UC}(n,p\times q)$ yields a $(p,q)$-approximation of Artin's classical braid group $B_n$. We describe a right-angled Artin group presentation for $B_n(p\times q)$ in cases where $\text{UC}(n,p\times q)$ is known to be aspherical. When $\min\{p,q\}=2$, our presentation agrees with Artin's classical presentation for $B_n$ removing the Artin-Tits relations. This allows us to deduce the value of the Lusternik-Schnirelmann category of the corresponding aspherical spaces $\text{UC}(n,p\times q)$, as well as the values of all their $k$-sequential topological complexities, both in the classical (Rudyak et al.) and distributional (Dransihnikov et al.) contexts. |
| title | A combinatorial genesis of the right-angled relations in Artin's classical braid groups |
| topic | Algebraic Topology 20F36, 55R80, 57M05, 57Q70 (20F65, 57M15, 82B26) |
| url | https://arxiv.org/abs/2504.12201 |