Generalized virtualization on welded links

Abstract

Let $n$ be a positive integer. The aim of this paper is to study two local moves $V(n)$ and $V^n$ on welded links, 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 which two welded links are equivalent up to $V^n$-moves. This leads 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.

18 pages, many figures; entirely revised, Section 7 added