Homotopy classification of ribbon tubes and welded string links

Abstract : 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.
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Annali della Scuola Normale Superiore di Pisa, 2017, XVII (5), pp.713-761
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  • HAL Id : hal-01017350, version 2
  • ARXIV : 1407.0184

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Benjamin Audoux, Paolo Bellingeri, Jean-Baptiste Meilhan, Emmanuel Wagner. Homotopy classification of ribbon tubes and welded string links. Annali della Scuola Normale Superiore di Pisa, 2017, XVII (5), pp.713-761. 〈hal-01017350v2〉

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