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 ${\alpha }_1\ne {\alpha }_2$ 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in the case where ${\alpha }_1,{\alpha }_2$ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author.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 ${\alpha }_1\ne {\alpha }_2$ 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.

We added a Theorem on comuting the minimal number of self intersection points using the Andersen-Mattes-Reshetikhin Poisson bracket. 20 pages, 5 figures

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