Intersections of Loops and the Andersen–Mattes–Reshetikhin Algebra

Authors

Patricia Cahn, University of PennsylvaniaFollow
Vladimir Chernov, Dartmouth College

Document Type

Article

Publication Date

1-2013

Publication Title

London Mathematical Society

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result of this paper in the case where α₁,α₂ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author.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.

Volume

87

Issue

3

First Page

785

Last Page

801

DOI

10.1112/jlms/jds065

Comments

Peer reviewed accepted manuscript.

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