Unified invariant of knots from homological braid action on Verma modules

Abstract

AbstractWe re‐build the quantum unified invariant of knots 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 . 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 nicely generalizing the well‐known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum non‐semisimple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant.