Welded Graphs, Wirtinger Groups and Knotted Punctured Spheres

Benjamin Audoux (), Jean-Baptiste Meilhan () and Akira Yasuhara ()
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Benjamin Audoux: Centrale Marseille, I2M, Aix Marseille Univ, CNRS
Jean-Baptiste Meilhan: Institut Fourier, Univ. Grenoble Alpes, CNRS
Akira Yasuhara: Faculty of Commerce, Waseda University

Chapter Chapter 18 in Essays on Topology, 2025, pp 389-427 from Springer

Abstract: Abstract 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; a group admitting such a presentation arises 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 $$\Upsilon $$ moves, under which the (non extended) Tube map is invariant.

Keywords: Surface-links in 4-space; Welded knotted objects; Gauss diagrams; Wirtinger group; Tube map; 57K45; 57K12; 20F05; 20F34; 57M15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/978-3-031-81414-3_18

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