Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra

TL;DR

This paper establishes that the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket equals the minimal intersection number of two distinct free homotopy classes of loops on a surface, revealing a precise algebraic-topological correspondence.

Contribution

It proves a new equality between Poisson bracket terms and minimal intersection numbers for loop classes, extending previous results and employing advanced algebraic techniques.

Findings

Number of Poisson bracket terms equals minimal intersection points for distinct classes.

The result does not hold for the Goldman Lie bracket in general.

Techniques from Turaev's cobracket study are used in the proof.

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paper in the case where $\alpha_1, \alpha_2$ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author.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