Unified invariant of knots from homological braid action on Verma modules

Abstract

We re-build the quantum sl2 unified invariant of knots $F_{\infty}$ 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, i.e. semi-simple and non semi-simple invariants of knots constructed from quantum sl2. We prove this last fact in our context which re-proves (a generalization of) the famous Melvin-Morton-Rozansky conjecture first proved by Bar-Natan and Garoufalidis. We find a symmetry of $F_{\infty}$ nicely generalizing the well known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum sl2 non semi-simple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express $F_{\infty}$ as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for $F_{\infty}$ using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant.