A generalization of Turaev's cobracket and the minimal self-intersection number of curve on a surface and a virtual string

Turaev introduced a Lie cobracket $D$ on the free $\mathbb{Z}$-module generated by the set of free homotopy classes of loops on a surface. Turaev conjectured that $D(A)=0$ if and only if $A$ is a power of a simple class. Chas constructed examples which show that Turaev's conjecture is unfortunately false. We construct an operation $M$ in the spirit of the Andersen-Mattes-Reshetikhin Poisson algebra of chord diagrams. The operation $M$ can be viewed as a generalization of Turaev's cobracket $D$. We show that Turaev's conjecture holds for $M$. Both the operations $M$ and $D$ give lower bounds on the minimum number of self-intersection points in a free homotopy class. Chas's examples show that the bound given by $D$ is not an equality in general. We show that the lower bound given by $M$ is an equality, so $M$ gives an explicit formula for the minimal self-intersection number. Finally, we consider the corresponding question for virtual strings. Turaev extended his cobracket $D$ to virtual strings, and $M$ extends to virtual strings as well. We show that the bound on the minimal self-intersection number of a virtual string given by $M$ is stronger than the bound given by $D$. We also show that the bound given by $M$ is at least as good as a bound given by Turaev's based matrix invariant.