A Dehornoy-Type Ordering on Plat Presentation Classes
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910115398942720 |
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| author | Ozawa, Makoto |
| author_facet | Ozawa, Makoto |
| contents | For each integer $n\ge 1$, after fixing a proper complexity function on the braid group $\B_{2n}$, we use the Dehornoy order to define a strict total order on the set \[ \mathcal P_{2n}=H_{2n}\backslash \B_{2n}/H_{2n} \] of $2n$--plat presentation classes. For a link type $\mathcal L$ with bridge number $b(\mathcal L)\le n$, this induces a strict total order on the subset $\mathcal P^{(n)}(\mathcal L)$ corresponding to bridge isotopy classes of $n$--bridge positions of $\mathcal L$. We also define a distinguished class $\CanPlat_D^{(n)}(\mathcal L)$ and show that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. As an application, we reformulate the fixed-level bridge finiteness conjecture in terms of boundedness of canonical representatives. This viewpoint supports the role of bridge positions as a structured finite-level model for studying the otherwise vast collection of geometric positions of a link. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07790 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Dehornoy-Type Ordering on Plat Presentation Classes Ozawa, Makoto Geometric Topology For each integer $n\ge 1$, after fixing a proper complexity function on the braid group $\B_{2n}$, we use the Dehornoy order to define a strict total order on the set \[ \mathcal P_{2n}=H_{2n}\backslash \B_{2n}/H_{2n} \] of $2n$--plat presentation classes. For a link type $\mathcal L$ with bridge number $b(\mathcal L)\le n$, this induces a strict total order on the subset $\mathcal P^{(n)}(\mathcal L)$ corresponding to bridge isotopy classes of $n$--bridge positions of $\mathcal L$. We also define a distinguished class $\CanPlat_D^{(n)}(\mathcal L)$ and show that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. As an application, we reformulate the fixed-level bridge finiteness conjecture in terms of boundedness of canonical representatives. This viewpoint supports the role of bridge positions as a structured finite-level model for studying the otherwise vast collection of geometric positions of a link. |
| title | A Dehornoy-Type Ordering on Plat Presentation Classes |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2604.07790 |