Searching for non-order-preserving braids algorithmically
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914971688894464 |
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| author | Johnson, Jonathan Scherich, Nancy Turner, Hannah |
| author_facet | Johnson, Jonathan Scherich, Nancy Turner, Hannah |
| contents | An $n$-strand braid is order-preserving if its action on the free group $F_n$ preserves some bi-order of $F_n$. A braid $β$ is order-preserving if and only if the link $L$ obtained as the union of the closure of $β$ and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid $β$, confirms this property and returns a proof that $β$ is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids $σ_1σ_2^{2m+1}$ are not order-preserving for any integer $m$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Searching for non-order-preserving braids algorithmically Johnson, Jonathan Scherich, Nancy Turner, Hannah Geometric Topology Group Theory 57K10, 57K20 An $n$-strand braid is order-preserving if its action on the free group $F_n$ preserves some bi-order of $F_n$. A braid $β$ is order-preserving if and only if the link $L$ obtained as the union of the closure of $β$ and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid $β$, confirms this property and returns a proof that $β$ is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids $σ_1σ_2^{2m+1}$ are not order-preserving for any integer $m$. |
| title | Searching for non-order-preserving braids algorithmically |
| topic | Geometric Topology Group Theory 57K10, 57K20 |
| url | https://arxiv.org/abs/2410.10595 |