Minimizing intersection points of loops on a surface and the Andersen–Mattes–Reshetikhin Poisson bracket (по совместной работе с Patricia Cahn)

Given two free homotopy classes $\alpha_1$, $\alpha_2$ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points $m(\alpha_1,\alpha_2)$ of loops in these two classes. We show that for $\alpha_1\neq\alpha_2$ the number of terms in the Andersen–Mattes–Reshetikhin Poisson bracket of $\alpha_1$ and $\alpha_2$ is equal to $m(\alpha_1,\alpha_2)$. Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of $\alpha_1$ and $\alpha_2$.
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 Turaev operation so that the conjecture is true for the modified operation.