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

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.