Goldman and Turaev constructed a Lie bialgebra structure on the free 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.