Parabolic skein modules

TL;DR

This paper develops a skein theory for 3-manifolds with surface defects, connecting quantum group representations to quantum character stacks and providing a new computational approach for knot invariants related to the quantum A-polynomial.

Contribution

It introduces a novel skein-theoretic framework for 3-manifolds with surface defects, extending quantum character stacks and offering a practical method for computing related knot invariants.

Findings

Extended skein theory to include surface defects in 3-manifolds.

Connected quantum group defects to quantum character stacks.

Provided a concrete computational method for quantum A-polynomial related invariants.

Abstract

We develop skein theory for 3-manifolds in the presence of codimension-one defects, focusing especially on defects arising from parabolic induction/restriction for quantum groups. We use these defects as a model for the quantum decorated character stacks of arXiv:2102.12283, thus extending them to 3-manifolds with surface defects. As a special case we obtain knot invariants closely related to the ``quantum $A$-polynomial", and we give a concrete method for computation resembling the approach of Dimofte and collaborators based on ideal triangulations and gluing equations.