Vassiliev invariants for virtual links, curves on surfaces and the Jones-Kauffman polynomial

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones-Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q, q(-1)]. We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants. It is also proved that this invariant is strictly stronger than the Jones-Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.