arXiv subjects

Patricia Cahn

Publications and source records attributed to Patricia Cahn.

20 records · Page 2Linked to original sources

arXiv2012

Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra

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 $m(α_1, α_2)$ of loops in these two classes. We show that for $α_1\neqα_2$ the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of $α_1$ and $α_2$ is equal to $m(α_1, α_2)$. 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of this paper in the case where $α_1, α_2$ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author.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 $α_1\neq α_2$ 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.

math.GT↗

arXiv2010

A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface

Goldman and Turaev constructed a Lie bialgebra structure on the free $\mathbb{Z}$-module generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket $Δ(α)$ is zero if and only if $α$ is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunately, false. We define an operation $μ$ in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through $μ$, so we can view $μ$ as a generalization of $Δ$. We show that Turaev's conjecture holds when $Δ$ is replaced with $μ$. We also show that $μ(α)$ gives an explicit formula for the minimum number of self-intersection points of a loop in $α$. The operation $μ$ also satisfies identities similar to the co-Jacobi and coskew symmetry identities, so while $μ$ is not a cobracket, $μ$ behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poisson algebra.

math.GT↗