Generalized virtualization on welded links

TL;DR

This paper introduces generalized local moves on welded links, demonstrating that one move unknots all welded knots while the other does not, and provides classifications and conditions for link equivalences.

Contribution

It defines and analyzes the properties of the generalized moves V(n) and V^n on welded links, including their roles as unknotting operations and their relations to link invariants.

Findings

V(n)-move is an unknotting operation for welded knots.

V^n-move is not an unknotting operation except for n=1.

Provides classification of welded links up to V(n)-moves.

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.