The Milnor $\bar{\mu}$ invariants and nanophrases

TL;DR

This paper extends Milnor's  invariants to nanophrases, a generalization of links, and explores their behavior under link homotopy in the context of virtual links.

Contribution

It introduces a generalization of  invariants to nanophrases and extends the concept of link homotopy to this broader setting.

Findings

Extended  invariants to nanophrases.

Generalized link homotopy to nanophrases.

Applied invariants to virtual links.

Abstract

Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self crossing changes. Milnor introduced invariants under link homotopy called $\bar{\mu}$. Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanophrases. We also generalize $\bar{\mu}$ to the set of those nanophrases that correspond to virtual links.