Homotopy classification of nanophrases with less than or equal to four letters

In this paper we give the stable classification of ordered, pointed, oriented multi-component curves on surfaces with minimal crossing number less than or equal to 2 such that any equivalent curve has no simply closed curves in its components. To do this, we use the theory of words and phrases which was introduced by V. Turaev. Indeed we give the homotopy classification of nanophrases with less than or equal to 4 letters. It is an extension of the classification of nanophrases of length 2 with less than or equal to 4 letters which was given by the author in a previous paper. This is a corrected version of Hokkaido University Preprint Series in Mathematics #921. I corrected the subsection 5.3 and added proofs of propositions.