k-reduced groups and Milnor invariants

TL;DR

This paper provides algebraic and diagrammatic characterizations of Milnor string link invariants with bounded index repetitions, using $k$-reduced free groups and welded knot theory, applicable to links as well.

Contribution

It introduces a novel algebraic and diagrammatic framework for Milnor invariants indexed by sequences with limited repetitions, extending understanding of link invariants.

Findings

Characterization of Milnor invariants via $k$-reduced free groups

Diagrammatic description using welded knot theory

Application to link cases

Abstract

We characterize, in an algebraic and in a diagrammatic way, Milnor string link invariants indexed by sequences where any index appears at most $k$ times, for any fixed $k\ge 1$. The algebraic characterization is given in terms of an Artin-like action on the so-called $k$-reduced free groups; the diagrammatic characterization uses the langage of welded knot theory. The link case is also addressed.