Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914716936306688 |
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| author | Dehornoy, Pierre Lunel, Corentin de Mesmay, Arnaud |
| author_facet | Dehornoy, Pierre Lunel, Corentin de Mesmay, Arnaud |
| contents | While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_09094 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect Dehornoy, Pierre Lunel, Corentin de Mesmay, Arnaud Computational Geometry Geometric Topology While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order. |
| title | Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect |
| topic | Computational Geometry Geometric Topology |
| url | https://arxiv.org/abs/2312.09094 |