Unified invariant of knots from homological braid action on Verma modules

TL;DR

This paper constructs a unified quantum invariant of knots from braid group actions on Verma modules, interpolating between colored Jones and ADO polynomials, and explores its symmetries and homological interpretations.

Contribution

It introduces a new two-variable knot invariant from quantum sl2 that unifies semi-simple and non semi-simple invariants, generalizes the Melvin-Morton-Rozansky conjecture, and provides a homological and determinant-based formula.

Findings

The invariant interpolates colored Jones and ADO polynomials.

It proves the Melvin-Morton-Rozansky conjecture in this context.

The invariant exhibits a symmetry similar to the Alexander polynomial.

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.