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

## TL;DR

This paper extends the classification of welded string links using Milnor invariants, demonstrating that certain moves and virtualizations can be characterized by these invariants, thus advancing understanding in knot theory.

## Contribution

It provides two new classifications of welded string links up to specific moves and virtualizations using Milnor invariants, generalizing previous results for classical links.

## Key findings

- Welded string links are classified up to $2n$-move and virtualization.
- Welded string links are classified up to $V^{n}$-move and virtualization.
- Milnor invariants effectively distinguish welded string links under these moves.

## Abstract

In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.