On the Goussarov-Polyak-Viro Finite-Type Invariants and the Virtualization Move

TL;DR

This paper proves that no nonconstant Goussarov-Polyak-Viro finite-type invariants remain invariant under the virtualization move, implying that certain polynomial invariants like the Jones-Kauffman polynomial are not of GPV finite type.

Contribution

It establishes a fundamental limitation of GPV finite-type invariants concerning virtualization invariance, clarifying their scope and properties.

Findings

No nonconstant GPV finite-type invariants are virtualization invariant.

Birman coefficients of the Jones-Kauffman polynomial are not GPV finite type.

The result constrains the classification of finite-type invariants.

Abstract

In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.