  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 R\'enyi 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 R\'enyi 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 R\'enyi 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
$n$.
