Twist Lattices and the Jones-Kauffman Polynomial for Long Virtual Knots

Details Twist Lattices and the Jones-Kauffman Polynomial for Long Virtual Knots Twist Lattices and the Jones-Kauffman Polynomial for Long Virtual Knots

2009-08-11

arxiv.org/abs/0908.1538v1

Mathematics

Geometric Topology

Scientific paper

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 $\le n$ if and only if an invariant is a polynomial of degree $\le n$ 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, where it is known for degree $k=2$.

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