Unified invariant of knots from homological braid action on Verma modules

We re‐build the quantum sl(2)${\mathfrak {sl}(2)}$ unified invariant of knots F∞$F_{\infty }$ from braid groups' action on tensors of Verma modules. It is a two variables series having the particularity of interpolating both families of colored Jones polynomials and ADO polynomials, that is, semisimple and non‐semisimple invariants of knots constructed from quantum sl(2)${\mathfrak {sl}(2)}$. We prove this last fact in our context that re‐proves (a generalization of) the famous Melvin–Morton–Rozansky conjecture first proved by Bar‐Natan and Garoufalidis. We find a symmetry of F∞$F_{\infty }$ nicely generalizing the well‐known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum sl(2)${\mathfrak {sl}(2)}$ non‐semisimple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express F∞$F_{\infty }$ as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for F∞$F_{\infty }$ using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant.