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We present a novel symmetry of the colored HOMFLY polynomial. It relatespairs of polynomials colored by different representations at specific values of$N$ and generalizes the previously known "tug-the-hook" symmetry of the coloredAlexander polynomial. As we show, the symmetry has a superalgebra origin, whichwe discuss qualitatively. Our main focus are the constraints that such aproperty imposes on the general group-theoretical structure, namely the$\mathfrak{sl}(N)$ weight system, arising in the perturbative expansion of theinvariant. Finally, we demonstrate its tight relation to the eigenvalueconjecture.

A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras