A diagrammatical characterization of Milnor invariants

TL;DR

This paper provides a diagrammatic method using welded knot theory to characterize Milnor invariants of links and string links, clarifying when these invariants are trivial or equal up to a certain length.

Contribution

It introduces a diagrammatic characterization of Milnor invariants using arrow calculus and $w_q$--concordance, connecting knot theory with link invariants.

Findings

Characterizes when two string links have equal Milnor invariants of length ≤ q

Determines when a link has trivial Milnor invariants of length ≤ q

Uses welded knot theory and arrow calculus for classification

Abstract

The goal of this paper is to give a diagrammatical characterization of the information given by the Milnor invariants of links and string links. More precisely, we describe when two string links have equal Milnor invariants of length $\leq q$ and when a link has trivial Milnor invariants of lenght $\leq q$. This will be done through the use of welded knot theory, involving the notions of arrow calculus and $w_q$--concordance introduced by J-B. Meilhan and A. Yasuhara. These results is to be compared to the classification of links up to $C_q$--concordance obtained by J. Conant, R. Schneiderman and P. Teichner.