Welded graphs, Wirtinger groups and knotted punctured spheres
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912503062069248 |
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| author | Audoux, Benjamin Meilhan, Jean-Baptiste Yasuhara, Akira |
| author_facet | Audoux, Benjamin Meilhan, Jean-Baptiste Yasuhara, Akira |
| contents | We develop a general diagrammatic theory of welded graphs, and provide an extension of Satoh's Tube map from welded graphs to ribbon surface-links. As a topological application, we obtain a complete link-homotopy classification of so-called knotted punctured spheres in $4$-space, by means of the $4$-dimensional Milnor invariants introduced previously by the authors. On the algebraic side, we show that the theory of welded graphs can be reinterpreted as a theory of Wirtinger group presentations, up to a natural set of transformations; these groups arise as the fundamental group of the exterior of the surface-link obtained from the given welded graph by the extended Tube map. Finally, we address the injectivity question for the Tube map, identifying a new family of local moves on welded links, called $Υ$ moves, under which the (non extended) Tube map is invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01922 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Welded graphs, Wirtinger groups and knotted punctured spheres Audoux, Benjamin Meilhan, Jean-Baptiste Yasuhara, Akira Geometric Topology 57K12, 57K45, 57M05 We develop a general diagrammatic theory of welded graphs, and provide an extension of Satoh's Tube map from welded graphs to ribbon surface-links. As a topological application, we obtain a complete link-homotopy classification of so-called knotted punctured spheres in $4$-space, by means of the $4$-dimensional Milnor invariants introduced previously by the authors. On the algebraic side, we show that the theory of welded graphs can be reinterpreted as a theory of Wirtinger group presentations, up to a natural set of transformations; these groups arise as the fundamental group of the exterior of the surface-link obtained from the given welded graph by the extended Tube map. Finally, we address the injectivity question for the Tube map, identifying a new family of local moves on welded links, called $Υ$ moves, under which the (non extended) Tube map is invariant. |
| title | Welded graphs, Wirtinger groups and knotted punctured spheres |
| topic | Geometric Topology 57K12, 57K45, 57M05 |
| url | https://arxiv.org/abs/2311.01922 |