Bridge trisections and Seifert solids
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| Soggetti: | |
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| _version_ | 1866918077114875904 |
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| author | Joseph, Jason Meier, Jeffrey Miller, Maggie Zupan, Alexander |
| author_facet | Joseph, Jason Meier, Jeffrey Miller, Maggie Zupan, Alexander |
| contents | We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard diagram. The Seifert solids produced can be assumed to have exteriors that can be built without 3-handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many 3-handles. We conclude with two classification results. The first shows that surfaces admitting doubly-standard shadow diagrams are unknotted. The second says that a $b$-bridge trisection in which some sector contains at least $b-1$ patches is completely decomposable, thus the corresponding surface is unknotted. This settles affirmatively a conjecture of the second and fourth authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09669 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bridge trisections and Seifert solids Joseph, Jason Meier, Jeffrey Miller, Maggie Zupan, Alexander Geometric Topology We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard diagram. The Seifert solids produced can be assumed to have exteriors that can be built without 3-handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many 3-handles. We conclude with two classification results. The first shows that surfaces admitting doubly-standard shadow diagrams are unknotted. The second says that a $b$-bridge trisection in which some sector contains at least $b-1$ patches is completely decomposable, thus the corresponding surface is unknotted. This settles affirmatively a conjecture of the second and fourth authors. |
| title | Bridge trisections and Seifert solids |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2210.09669 |