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.