Connected sum of virtual knots and F -polynomials

It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman's affine index polynomial. For any pair of virtual knots K and M with n-dwrithe δJn(K)=0 we construct an infinite family of different connected sums of K and M which can be distinguished by F-polynomials.