Generalized virtualization on welded links

Milnor invariants, $2n$-moves and $V^{n}$-moves for welded string links

“…On the other hand, surgery along a B-move is equivalent to a V n -move. Furthermore, it is not hard to see that a B-move is equivalent to a C-move; see [21].…”

“…Here, the (2n + sv)-equivalence is the equivalence relation on welded string links generated by the 2n-move, self-crossing virtualization and welded isotopy. Here a V n -move, defined by the authors in [21], is an oriented local move as illustrated in Figure 1.2 which is a generalization of the crossing virtualization. (In fact, the V 1 -move is equivalent to the crossing virtualization.)…”