A Generalization of Turaev's Virtual String Cobracket

In a previous paper, we defined an operation $\mu$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings. We show that $\mu$ gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev's virtual string cobracket. We use Turaev's based matrices to describe strings $\alpha$ such that $\mu$ gives a formula for the minimal self-intersection number $\alpha$. We also construct an example that shows the bound on the minimal self-intersection number given by $\mu$ is sometimes stronger than the bound $\rho$ given by Turaev's based matrix invariant.