#### Document Type

Article

#### Publication Date

1-2013

#### Publication Title

London Mathematical Society

#### Abstract

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 # (α_{1},α_{2}) of loops in these two classes.

We show that, for α_{1}≠α_{2}, the number of terms in the Andersen–Mattes–Reshetikhin Poisson bracket of α_{1} and α_{2} is equal to # (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of this paper in the case where α_{1},α_{2} do *not* contain different powers of the same loop first appeared in the unpublished preprint of the second author.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}≠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.

#### Volume

87

#### Issue

3

#### First Page

785

#### Last Page

801

#### DOI

10.1112/jlms/jds065

#### Recommended Citation

Cahn, Patricia and Chernov, Vladimir, "Intersections of Loops and the Andersen–Mattes–Reshetikhin Algebra" (2013). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.

https://scholarworks.smith.edu/mth_facpubs/57

## Comments

Peer reviewed accepted manuscript.