TWIST LATTICES AND THE JONES–KAUFFMAN POLYNOMIAL FOR LONG VIRTUAL KNOTS

Abstract

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.