In this talk, we consider ribbon tubes, which are knotted annuli in 4-space bounding ribbon 3-balls. We will see how ribbon tubes naturally act on the reduced free group, and that this action classifies ribbon tubes up to homotopy, that is, when allowing each tube component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization, and the above-mentionned action on the reduced free group can be seen as the “virtual extension” of Milnor invariants. This generalizes a result of Habegger and Lin on string links.
This talk is based on a joint work with B. Audoux, J-B. Meilhan and E. Wagner.