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 $k\geq 3$.Comment: 13 pages, 6 figures; v2: appendix added; v3:rank formula corrected;
to appear in Journal of Topolog
In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a by-product, we obtain that all of the above local moves are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.
We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or selfvirtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.• P n and SL n stand for the (usual) sets of pure braids and string links on n strands, and the prefix v and w refer to their virtual and welded counterpart, respectively; • the superscripts sv and sc refer respectively to the equivalence relations generated by self-virtualization and self-crossing change.
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