A program to construct (pre)modular tensor categories from 3-manifolds, first started by Cho-Gang-Kim using the $M$ theory of physics and studied mathematically by Cui-Qiu-Wang continue. A relevant and important construct is a collection of a given $\text{SL}(2, \mathbb{C})$ character on a given manifold, which serves as a simple object type for the corresponding category. Chern-Simons invariants and adjoint Leidemeister twists play important roles in construction and are related to topological twists and quantum dimensions of simple objects, respectively. The modular $S$ matrix is computed from local operators, following a trial-and-error procedure. It is currently unknown how to generate data beyond the modular $S$- and $T$- matrices. Also, there are many subtleties in the structure that have yet to be resolved. In this paper, we consider an infinite family of 3-manifolds, the torus bundle on a circle. We show that the modular data produced by such manifolds are realized by $\mathbb{Z}_2$ idempotent variations of a particular pointed premodular category. Here an equal change is performed on the $\mathbb{Z}_2$ action and a simple (flippable) object sent to the other way around. This is also called particle Hall symmetry. I hope this extensive class example sheds light on how the program can be improved to recover the full data of the categories before modularization.
