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 $\Delta(\alpha)$ is zero if and only if $\alpha$ is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunately, false. We define an operation $\mu$ in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through $\mu$, so we can view $\mu$ as a generalization of $\Delta$. We show that Turaev's conjecture holds when $\Delta$ is replaced with $\mu$. We also show that $\mu(\alpha)$ gives an explicit formula for the minimum number of self-intersection points of a loop in $\alpha$. The operation $\mu$ also satisfies identities similar to the co-Jacobi and coskew symmetry identities, so while $\mu$ is not a cobracket, $\mu$ behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poisson algebra.