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.

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