Narrow Results

In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.

In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov–Polyak–Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree ≤ n if and only if an invariant is a polynomial of degree ≤ n on every twist lattice of the right form. The main result of this paper is an application of this technique to the coefficients of the Jones–Kauffman polynomial. It is shown that the Kauffman finite-type invariants obtained from these coefficients are not GPV finite-type invariants of any degree by explicitly showing they can never be polynomials. This generalizes a result of Kauffman [8], where it is known for degree k = 2.