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

TL;DR

This paper explores the relationship between twist sequences and finite-type invariants for long virtual knots, demonstrating that certain Jones-Kauffman polynomial coefficients are not of finite type in the Goussarov-Polyak-Viro sense.

Contribution

It introduces a novel application of twist sequences to analyze the finite-type properties of Jones-Kauffman polynomial coefficients for long virtual knots.

Findings

Jones-Kauffman polynomial coefficients are not GPV finite-type invariants of any degree.

Finite-type invariants of degree ≤ n correspond to polynomial behavior on specific twist lattices.

Generalizes Kauffman's degree 2 result to all degrees.

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 $\le n$ 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 $k=2$.