Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra

Given two free homotopy classes $α_1, α_2$ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points $m(α_1, α_2)$ of loops in these two classes. We show that for $α_1\neqα_2$ the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of $α_1$ and $α_2$ is equal to $m(α_1, α_2)$. Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of $α_1$ and $α_2$. The main result of this paper in the case where $α_1, α_2$ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author. In order to prove the main result for all pairs of $α_1\neq α_2$ we had to use the techniques developed by the first author in her study of operations generalizing Turaev's cobracket of loops on a surface.

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