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Pré-Publication, Document De Travail Année : 2023

Welded graphs, Wirtinger groups and knotted punctured spheres

Résumé

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 \emph{knotted punctured spheres} in 4-space, by means of the 4-dimensional Milnor invariants introduced by the authors in a previous paper. 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 $\Upsilon$ moves, under which the (non extended) Tube map is invariant.
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Dates et versions

hal-04272362 , version 1 (06-11-2023)

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Benjamin Audoux, Jean-Baptiste Meilhan, Akira Yasuhara. Welded graphs, Wirtinger groups and knotted punctured spheres. 2023. ⟨hal-04272362⟩
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