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

Understand

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.