Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups

We study the entanglement for a state on linked torus boundaries in $3d$ Chern-Simons theory with a generic gauge group and present the asymptotic bounds of Renyi entropy at two different limits: (i) large Chern-Simons coupling $k$, and (ii) large rank $r$ of the gauge group. These results show that the Renyi entropies cannot diverge faster than $ln k$ and $ln r$, respectively. We focus on torus links $T(2,2n)$ with topological linking number $n$. The Renyi entropy for these links shows a periodic structure in $n$ and vanishes whenever $n = 0 text{ (mod } textsf{p})$, where the integer $textsf{p}$ is a function of coupling $k$ and rank $r$. We highlight that the refined Chern-Simons link invariants can remove such a periodic structure in $n$.