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In the study of 3d-3d correspondence occurs a natural $q$-Weyl algebraassociated to an ideal triangulation of a 3-manifold with torus boundarycomponents, and a module of it. We study the action of this module on the(rotated) 3d-index of Dimofte-Gaiotto-Gukov and we conjecture some structuralproperties: bilinear factorization in terms of holomorphic blocks, pair oflinear $q$-difference equations, the determination of the 3d-index in terms ofa finite size matrix of rational functions and the asymptotic expansion of the$q$-series as $q$ tends to 1 to all orders. We illustrate our conjectures withcomputations for the case of the three simplest hyperbolic knots.

The descendants of the 3d-index