Combinatorial approach to Milnor invariants of welded links

TL;DR

This paper introduces a combinatorial method to extend Milnor invariants to welded links, providing an alternative proof of their invariance and broadening their applicability in knot theory.

Contribution

It offers a novel combinatorial approach to defining Milnor invariants for welded links, complementing existing topological methods.

Findings

Successfully extends Milnor invariants to welded links

Provides an alternative proof of classical invariants' invariance

Enhances understanding of link invariants in welded link theory

Abstract

For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\overline{\mu}$-invariants. Recently, Chrisman extended Milnor $\overline{\mu}$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $\overline{\mu}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $\overline{\mu}$-invariants of classical links.