Differential hierarchy and additional grading of knot polynomials

TL;DR

This paper explores the structure of colored knot polynomials, revealing a differential hierarchy and new coordinate systems that clarify their properties without introducing new invariants.

Contribution

It demonstrates that the recently proposed fourth grading is a redefinition of existing coordinates, highlighting the underlying differential hierarchy in knot polynomials.

Findings

The Z-expansion encodes knot polynomial coefficients as functions of (A,q,t).

The new coordinates simplify when representations are embedded into fundamental products.

The fourth grading is a redefinition, not a new invariant.

Abstract

Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of alternative character expansion. These new variables are decomposed in an especially simple way, when the representation is embedded into a product of the fundamental ones. The fourth grading recently proposed in arXiv:1304.3481, seems to be just a simple redefinition of these new coordinates, elegant but in no way distinguished. If so, it does not provide any new independent knot invariants, instead it can be considered as one more testimony of the hidden differential hierarchy (Z-expansion) structure behind the knot polynomials.