标题： 焊接图，维特林群和扭结穿孔球面 显示英文标题

标题： Welded graphs, Wirtinger groups and knotted punctured spheres

摘要： 我们发展了一种焊接图的一般图示理论，并将Satoh的Tube映射从焊接图扩展到带状曲面链环。 作为拓扑应用，我们通过作者在\cite{cutAMY}中引入的$4$维Milnor不变量，得到了所谓的\emph{结状穿孔球体}在$4$空间中的完整链环同伦分类。 在代数方面，我们证明了焊接图的理论可以重新解释为Wirtinger群表示的理论，直到一个自然的变换集；这些群作为由扩展Tube映射从给定焊接图得到的曲面链环的外部的基本群而出现。 最后，我们研究了Tube映射的单射性问题，确定了一个新的焊接链环上的局部移动族，称为$\Upsilon$移动，在这些移动下（非扩展）Tube映射是不变的。 显示英文摘要

摘要： 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 \cite{cutAMY}. 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.