Minimizing intersection points of loops on a surface and the Andersen-Mattes-Reshetikhin Poisson bracket

(based on a joint work with Patricia Cahn)
Given two free homotopy classes α1,α2 of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points m(α1,α2) of loops in these two classes. We show that for α1≠α2 the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of α1 and α2 is equal to m(α1,α2). Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of α1 and α2.
If time permits we will also discuss the following. Turaev conjectured that his cobracket of a loop is zero if and only if the loop is homotopic to a simple loop. Counterexamples to this conjecture were found by Chas. Cahn modified the Turae'v operation so that the conjecture is true for the modified operation.