  We construct the infinite sequence of invariants for curves in surfaces by
using word theory that V. Turaev introduced. For plane closed curves, we add
some extra terms, e.g. the rotation number. From these modified invariants, we
get the Arnold's basic invariants and some other invariants. We also express
how these invariants classify plane closed curves. In addition, we consider
other classes of plane curves: long curves and fronts.
