Homotopy classification of ribbon tubes and welded string links

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon tubes, which are knotted annuli bounding ribbon 3-balls. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-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. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general "virtual extension" of Milnor invariants. We also give a classification of ribbon torus-links up to link-homotopy. Finally, connections between usual, virtual and welded knotted objects are investigated.