HOMOTOPY CLASSIFICATION OF NANOPHRASES IN TURAEV'S THEORY OF WORDS.

The purpose of this paper is to give the homotopy classification of nanophrases of length 2 with 4 letters. To do it we construct some new invariants of nanophrases γ and T. The invariant γ defined in this paper is an extension of the invariant γ for nanowords introduced by Turaev. The invariant T is a new invariant of nanophrases. As a corollary of these results, we give the classification of two-component pointed, ordered, oriented curves on surfaces with minimum crossing number less than or equal to 2. [ABSTRACT FROM AUTHOR]