  Goldman and Turaev found a Lie bialgebra structure on the vector space
generated by non-trivial free homotopy classes of curves on a surface. When the
surface has non-empty boundary, this vector space has a basis of cyclic reduced
words in the generators of the fundamental group and their inverses. We give a
combinatorial algorithm to compute this Lie bialgebra on this vector space of
cyclic words. Using this presentation, we prove a generalization of Goldman's
result relating the bracket to disjointness of curve representatives when one
of the classes is simple. We exhibit some examples we found by programming the
algorithm which answer negatively Turaev's question about the characterization
of simple curves in terms of the cobracket. Further computations suggest an
alternative characterization of simple curves in terms of the bracket of a
curve and its inverse. Turaev's question is still open in genus zero.
