Large-$N$ Torus Knots in Lens Spaces and Their Quiver Structure

We study torus knot invariants in the lens space $S^{3}/\mathbb{Z}_{p}$ within Chern--Simons theory. Using the surgery and modular description of lens spaces, we derive a general expression for the invariant of an $(\alpha,\beta)$ torus knot in this background. In the large-$N$ limit these invariants simplify and acquire a universal form: the invariant of an $(\alpha,\beta)$ torus knot in $S^{3}/\mathbb{Z}_{p}$ can be expressed in terms of the invariant of the $(\alpha,\alpha+p\beta)$ torus knot in $S^{3}$. After an appropriate redefinition of knot variables, the generating functions of these invariants exhibit a structure analogous to quiver partition functions. Since the associated quiver is independent of the rank $N$ and level $k$ of Chern--Simons theory, the large-$N$ result provides a direct way to identify the underlying quiver, allowing us to determine the quiver structure associated with torus knots in $S^{3}/\mathbb{Z}_{p}$.