Bridge Multisections of Knotted Surfaces in $S^4$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917398721855488 |
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| author | Aranda, Román Engelhardt, Carolyn |
| author_facet | Aranda, Román Engelhardt, Carolyn |
| contents | Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we prove a uniqueness result for bridge multisections of surfaces in 4-space: we give a complete set of moves relating to any two multiplane diagrams of the same surface. This is done by developing a surgery operation on multiplane diagrams called band surgery. Another application of this surgery move is that any $n$-valent graph with an $n$-edge coloring is the spine of a bridge multisection for an unknotted surface. We also prove that any multisected surface in $S^4$ can be unknotted by finitely many band surgeries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bridge Multisections of Knotted Surfaces in $S^4$ Aranda, Román Engelhardt, Carolyn Geometric Topology 57Q45 Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we prove a uniqueness result for bridge multisections of surfaces in 4-space: we give a complete set of moves relating to any two multiplane diagrams of the same surface. This is done by developing a surgery operation on multiplane diagrams called band surgery. Another application of this surgery move is that any $n$-valent graph with an $n$-edge coloring is the spine of a bridge multisection for an unknotted surface. We also prove that any multisected surface in $S^4$ can be unknotted by finitely many band surgeries. |
| title | Bridge Multisections of Knotted Surfaces in $S^4$ |
| topic | Geometric Topology 57Q45 |
| url | https://arxiv.org/abs/2410.01921 |