The first-order deviation of superpolynomial in an arbitrary representation from the special polynomial

TL;DR

This paper proposes a simple formula for the first-order deviation of superpolynomials from the special polynomial in arbitrary representations, addressing a gap in the understanding of superpolynomials beyond symmetric or antisymmetric cases.

Contribution

It introduces a conjectural formula for the first-order deviation of superpolynomials in arbitrary representations, providing a new tool for testing superpolynomial calculations.

Findings

Proposes a simple, potentially universal formula for the first-order deviation.

Addresses the lack of examples of superpolynomials in arbitrary representations.

Provides a basis for future validation of superpolynomial formulas.

Abstract

Like all other knot polynomials, the superpolynomials should be defined in arbitrary representation R of the gauge group in (refined) Chern-Simons theory. However, not a single example is yet known of a superpolynomial beyond symmetric or antisymmetric representations. We consider the expansion of the superpolynomial around the special polynomial in powers of (q-1) and (t-1) and suggest a simple formula for the first-order deviation, which is presumably valid for arbitrary representation. This formula can serve as a crucial lacking test of various formulas for non-trivial superpolynomials, which will appear in the literature in the near future.