We consider the matrix model of U(N) refined Chern–Simons theory on S3\documentclass[12pt]{minimal}
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\begin{document}$$S^3$$\end{document} for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern–Simons for a gauge super-group U(N|M). Our equations and solutions are manifestly symmetric under Langlands duality q↔t-1\documentclass[12pt]{minimal}
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\begin{document}$$q\leftrightarrow t^{-1}$$\end{document} which correctly reproduces 3d Seiberg duality when q is a specific root of unity.
