Characterizations of knot groups and knot symmetric quandles of surface-links
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917933279608832 |
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| author | Yasuda, Jumpei |
| author_facet | Yasuda, Jumpei |
| contents | The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method. González-Acuña and Kamada independently extended this characterization to the knot groups of orientable surface-links. Kamada applied the closed 2-dimensional braid method to show this result. In this paper, we generalize these results to characterize the knot groups of surface-links, including non-orientable ones. We use a plat presentation for surface-links to prove it. Furthermore, we show a similar characterization for the knot symmetric quandles of surface-links. As an application, we show that every dihedral quandle with an arbitrarily good involution can be realized as the knot symmetric quandle of a surface-link. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizations of knot groups and knot symmetric quandles of surface-links Yasuda, Jumpei Geometric Topology The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method. González-Acuña and Kamada independently extended this characterization to the knot groups of orientable surface-links. Kamada applied the closed 2-dimensional braid method to show this result. In this paper, we generalize these results to characterize the knot groups of surface-links, including non-orientable ones. We use a plat presentation for surface-links to prove it. Furthermore, we show a similar characterization for the knot symmetric quandles of surface-links. As an application, we show that every dihedral quandle with an arbitrarily good involution can be realized as the knot symmetric quandle of a surface-link. |
| title | Characterizations of knot groups and knot symmetric quandles of surface-links |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2412.20081 |