Unified invariant of knots from homological braid action on Verma modules

Jules Martel; Sonny Willetts

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arXiv · 2112.15204

Unified invariant of knots from homological braid action on Verma modules

Jules Martel·Sonny Willetts

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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.

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BibTeX RIS

Jules Martel, Sonny Willetts. 2021-12-30. Unified invariant of knots from homological braid action on Verma modules. https://arxiv.org/abs/2112.15204

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