New York Journal of Mathematics
Goldman and Turaev constructed a Lie bialgebra structure on the free Zmodule generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket ∆(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 ∆. We show that Turaev’s conjecture holds when ∆ 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 AndersenMattes-Reshetikhin Poisson algebra.
Cahn, Patricia, "A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface" (2013). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.