Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots

In [9], Kauffman introduced virtual knot theory and generalized many classical knot invariants to virtual ones. For example, he extended the Jones polynomials V-K (t) of classical links to the f-polynomials f(K) (A) of virtual links by using bracket polynomials. In [4], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots. In this paper, we give a necessary condition for a virtual knot invariant to be of finite type by using t(a(1), . . . , a(m)) sequences of virtual knots. Then we show that the higher derivatives f(K)((n)) (a) of the f-polynomial f(K) (A) of a virtual knot K at any point a are not of finite type unless n < 1 and a = 1.