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Welded Graphs, Wirtinger Groups and Knotted Punctured Spheres

In: Essays on Topology

Author

Listed:
  • 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)

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.

Suggested Citation

  • Benjamin Audoux & Jean-Baptiste Meilhan & Akira Yasuhara, 2025. "Welded Graphs, Wirtinger Groups and Knotted Punctured Spheres," Springer Books, in: Louis Funar & Athanase Papadopoulos (ed.), Essays on Topology, chapter 0, pages 389-427, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-81414-3_18
    DOI: 10.1007/978-3-031-81414-3_18
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