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.
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.
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