Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra

Abstract

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 $\alpha_2$. The main result of this paper in the case where $\alpha_1, \alpha_2$ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author.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 $\alpha_1\neq \alpha_2$ 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.