Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots
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In [9], Kauffman introduced virtual knot theory and generalized many classical knot invariants to virtual ones. For example, he extended the Jones polynomials VK(t) of classical links to the f{polynomials fK(A) of virtual links by using bracket polynomials. In [4], M. Goussarov, M. Polyak and O. Viro introduced ¯nite 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(a1,ㆍㆍㆍ,am)-sequences of virtual knots. Then we show that the higher derivatives f(n)K(a) of the f-polynomial fK(A) of a virtual knot K at any point a are not of ¯nite type unless n ≤ 1 and a = 1.
Abstract
In [9], Kauffman introduced virtual knot theory and generalized many classical knot invariants to virtual ones. For example, he extended the Jones polynomials VK(t) of classical links to the f{polynomials fK(A) of virtual links by using bracket polynomials. In [4], M. Goussarov, M. Polyak and O. Viro introduced ¯nite 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(a1,ㆍㆍㆍ,am)-sequences of virtual knots. Then we show that the higher derivatives f(n)K(a) of the f-polynomial fK(A) of a virtual knot K at any point a are not of ¯nite type unless n ≤ 1 and a = 1.
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