Parabolic skein modules

TL;DR

The paper builds a defect-aware skein formalism for 3-manifolds, centering on parabolic defects and their relation to decorated character stacks, to define and compute a localised quantum A-ideal that q-deforms the classical A-polynomial. It introduces parabolic defect skeins, redecorated representations, and internal skein algebras, then develops quantum tori and cluster charts to organize computations via ideal triangulations. A central result expresses the localised A-ideal as an elimination of bulk and thread relations in a quantum torus, enabling finite-time computation with noncommutative Groebner methods and explicit examples for several knots, including Pachner moves. The framework provides invariant, triangulation-controlled knot invariants tied to the quantum A-polynomial, with potential applications to extended TQFTs and domain walls between quantum field theories.

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.

Figures (21)

• Figure 1: Coordinates defined by some typical loops in $M$.
• Figure 2: At left, a parabolic defect skein: here $V$ and $\chi$ are objects of $\mathop{\mathrm{Rep}}\nolimits_q G$ and $\mathop{\mathrm{Rep}}\nolimits_q T$, respectively, and $f\in \mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathrm{Rep}}\nolimits_qB}(\operatorname{Ind}_T^B(\chi),\mathop{\mathrm{Res}}\nolimits_B^G(V))$. At right, skeins in $M^\circ$ which quantize the functions from Figure \ref{['fig:functions-on-Ch']}.
• Figure 3: Additional skein relations satisfied at the parabolic defect. The first relation is a consequence of stratified isotopy invariance, while the latter three relations are easy computations with parabolic induction and restriction.
• Figure 4: Two depictions of the same decorated surface $\Sigma_{\triangle}$ associated to the figure-eight knot and an ideal triangulation. The surface $\Sigma_\triangle$ consists of a $T$-region torus, onto which one $G$-region handle is attached for each long edge of the triangulation. On the left, $\Sigma_\triangle$ is drawn emphasizing the shape of the knot as it sits in $S^3$. On the right, the same $\Sigma_\triangle$ is drawn to emphasize the role of the ideal triangulation. The eight punctures indicate the location of the tetrahedra's vertices, while the connecting cylinders show identified faces.
• Figure 5: Attaching $T$-coloured disks to the boundary of $M^{bulk}$.

Theorems & Definitions (78)

• Example 1.1
• Remark 1.2
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6: Theorem \ref{['thm:its-the-quantum-A-polynomial']}
• Remark 1.7
• Example 1.8: The trefoil
• Remark 1.9
• Example 1.10: The figure-eight