Intersections of loops and the Andersen-Mattes-Reshetikhin algebra

Abstract

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 # (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 # (α1,α2). Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of α1 and α2. The main result of this paper in the case where α1,α2 do not contain different powers of the same loop first appeared in the unpublished preprint of the second author.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 α1≠α2, 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.