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

摘要

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. Following the article Equations on knot polynomials and 3d/5d duality, 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.