Finiteness criteria for defect skein modules

Determine which braided module categories over \(\operatorname{Rep}_q(G)\) guarantee that the defect skein module of every closed 3-manifold with a defect knot is finite dimensional over \(\mathbb{C}(q)\).

Background

The paper establishes finite dimensionality for defect skein modules associated with the quantum symmetric-pair braided module categories, subject to a holonomicity assumption on the relevant modules of geometric origin. It also notes that finiteness fails for some braided module categories, such as the annular factorization-homology module. The broader classification problem of defect labels guaranteeing finiteness therefore remains unresolved.

References

Let $M$ be a closed 3-manifold with a defect knot $L\subset M$. What type of defect labels $M$ (braided module categories over $A = Rep_qG$) guarantee that the defect skein module $Sk_{(Rep_q(G),M )}(M,L)$ is finite dimensional over $C(q)$?

— Skein theory, line defects, and quantum symmetric pairs  (2609.18902 - Chen et al., 16 Sep 2026) in Question 1, Section 1, subsection “Codimension two defects from braided module categories”