Welded graphs, Wirtinger groups and knotted punctured spheres

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 $\Upsilon$ moves, under which the (non extended) Tube map is invariant.