Goldman and Turaev discovered a Lie bialgebra structure on the vector space generated by free homotopy classes of loops on an oriented surface. Goldman's Lie bracket gives a lower bound on the minimum number of intersection points of two loops in two given free homotopy classes. Turaev's Lie cobracket gives a lower bound on the minimum number of self- intersection points of a loop in a given free homotopy class. Chas showed that these bounds are not equalities in general. We show that for other operations, namely, the Andersen-Mattes-Reshetikhin Poisson bracket and a new operation "mu", the corresponding bounds are always equalities. Some of this is joint work with Vladimir Chernov.