On the classification of 2-plat 2-knots
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909008667869184 |
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| author | Yasuda, Jumpei |
| author_facet | Yasuda, Jumpei |
| contents | An $n$-plat 1-knot is one isotopic to the plat closure of some $2n$-braid, which is also called an $n$-bridge 1-knot. Schubert classified 2-bridge 1-knots by considering their double branched covers which are homeomorphic to lens spaces. A 2-knot is a 2-sphere smoothly embedded in 4-space or 4-sphere. An $n$-plat 2-knot is one isotopic to the plat closure of some 2-dimensional $2n$-braid. The aim of this paper is to classify 2-plat 2-knots. By a result of Montesinos, double branched covers do not distinguish 2-plat 2-knots. Thus, we introduce a new invariant to classify them. Our invariant serves as an analogue of a torsion invariant. Furthermore, it is an obstruction to invertibility of 2-knots. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_15401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the classification of 2-plat 2-knots Yasuda, Jumpei Geometric Topology An $n$-plat 1-knot is one isotopic to the plat closure of some $2n$-braid, which is also called an $n$-bridge 1-knot. Schubert classified 2-bridge 1-knots by considering their double branched covers which are homeomorphic to lens spaces. A 2-knot is a 2-sphere smoothly embedded in 4-space or 4-sphere. An $n$-plat 2-knot is one isotopic to the plat closure of some 2-dimensional $2n$-braid. The aim of this paper is to classify 2-plat 2-knots. By a result of Montesinos, double branched covers do not distinguish 2-plat 2-knots. Thus, we introduce a new invariant to classify them. Our invariant serves as an analogue of a torsion invariant. Furthermore, it is an obstruction to invertibility of 2-knots. |
| title | On the classification of 2-plat 2-knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2506.15401 |