On codimension two embeddings up to link-homotopy

Abstract

We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ribbon k-string links, for $k\ge 3$.

13 pages, 6 figures; v2: appendix added; v3:rank formula corrected; to appear in Journal of Topology

References in corpus (1)

• Arrow calculus for welded and classical links

Cited by in corpus (6)

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• On braids and links up to link-homotopy
• Milnor invariants, $2n$-moves and $V^n$-moves for welded string links
• Milnor-type invariants for surface-links and cut-diagrams