Generalized virtualization on welded links

Abstract

The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links for a positive integer $n$, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves. On the other hand, we give a necessary condition for two welded links to be equivalent up to $V^{n}$-moves. This leads us to show that the $V^{n}$-move is not an unknotting operation on welded knots except for $n = 1$. We also discuss relations among $V^{n}$-moves, associated core groups and the multiplexing of crossings.