  A flat virtual link is a finite collection of oriented closed curves
$\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy,
i.e., a composition of elementary stabilizations, destabilizations, and
homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the
minimal number of intersection points of curves in the virtual homotopy class
of $(L_1, L_2)$ equals to the number of terms of a generalization of the
Anderson--Mattes--Reshetikhin Poisson bracket. Furthermore, considering a
single curve, we show that the minimal number of self-intersections of a curve
in its virtual homotopy class can be counted by a generalization of the Cahn
cobracket.
