Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918520397234176 |
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| author | de Paiva, Thiago |
| author_facet | de Paiva, Thiago |
| contents | Lorenz links and T-links are equivalent families by the work of Birman--Kofman. Lorenz links arise as periodic orbits of the Lorenz system, whereas T-links are closures of certain positive braids. Birman, Williams, and Franks showed that every Lorenz link admits a positive braid with a full twist realizing the braid index. We study Lorenz links from the minimal-braid viewpoint. We compute these representatives from the T-link parameters and show that they are precisely a family of positive braids with a full twist, called V-link braids. This gives an explicit correspondence between V-links and T-links. We use this correspondence to study equivalences among T-link presentations. We obtain criteria for distinguishing T-links from their parameters, recover the Birman--Kofman equivalence from the V-link/T-link correspondence, and prove that, under mild assumptions, this equivalence gives two distinct presentations of the same link. We also generalize this phenomenon: under natural non-degeneracy conditions, one obtains four distinct T-link presentations of the same link. More generally, even with the braid index fixed, the number of distinct V-link and T-link presentations can be arbitrarily large. In contrast, we initiate the study of uniqueness of T-link presentations. We prove that there are infinite families of Lorenz links with a unique T-link presentation, after excluding the destabilization case. For the Lorenz flow, this gives families of periodic orbits that admit only one T-link parameter description in their isotopy class. Finally, the V-link form places Lorenz links in the setting of positive braids with a full twist, where techniques for detecting geometric type apply. We prove criteria for such braids to be satellite or hyperbolic and apply them to V-links and T-links. As a consequence, we obtain new families of satellite and hyperbolic T-links and generalize known results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types de Paiva, Thiago Geometric Topology 57K10, 57K31, 57K32, 37D45 Lorenz links and T-links are equivalent families by the work of Birman--Kofman. Lorenz links arise as periodic orbits of the Lorenz system, whereas T-links are closures of certain positive braids. Birman, Williams, and Franks showed that every Lorenz link admits a positive braid with a full twist realizing the braid index. We study Lorenz links from the minimal-braid viewpoint. We compute these representatives from the T-link parameters and show that they are precisely a family of positive braids with a full twist, called V-link braids. This gives an explicit correspondence between V-links and T-links. We use this correspondence to study equivalences among T-link presentations. We obtain criteria for distinguishing T-links from their parameters, recover the Birman--Kofman equivalence from the V-link/T-link correspondence, and prove that, under mild assumptions, this equivalence gives two distinct presentations of the same link. We also generalize this phenomenon: under natural non-degeneracy conditions, one obtains four distinct T-link presentations of the same link. More generally, even with the braid index fixed, the number of distinct V-link and T-link presentations can be arbitrarily large. In contrast, we initiate the study of uniqueness of T-link presentations. We prove that there are infinite families of Lorenz links with a unique T-link presentation, after excluding the destabilization case. For the Lorenz flow, this gives families of periodic orbits that admit only one T-link parameter description in their isotopy class. Finally, the V-link form places Lorenz links in the setting of positive braids with a full twist, where techniques for detecting geometric type apply. We prove criteria for such braids to be satellite or hyperbolic and apply them to V-links and T-links. As a consequence, we obtain new families of satellite and hyperbolic T-links and generalize known results. |
| title | Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types |
| topic | Geometric Topology 57K10, 57K31, 57K32, 37D45 |
| url | https://arxiv.org/abs/2409.14824 |