  We study the multi-boundary entanglement structure of the state associated
with the torus link complement $S^3 \backslash T_{p,q}$ in the set-up of
three-dimensional SU(2)$_k$ Chern-Simons theory. The focal point of this work
is the asymptotic behavior of the R\'enyi entropies, including the entanglement
entropy, in the semiclassical limit of $k \to \infty$. We present a detailed
analysis for several torus links and observe that the entropies converge to a
finite value in the semiclassical limit. We further propose that the large $k$
limiting value of the R\'enyi entropy of torus links of type $T_{p,pn}$ is the
sum of two parts: (i) the universal part which is independent of $n$, and (ii)
the non-universal or the linking part which explicitly depends on the linking
number $n$. Using the analytic techniques, we show that the universal part
comprises of Riemann zeta functions and can be written in terms of the
partition functions of two-dimensional topological Yang-Mills theory. More
precisely, it is equal to the R\'enyi entropy of certain states prepared in
topological $2d$ Yang-Mills theory with SU(2) gauge group. Further, the
universal parts appearing in the large $k$ limits of the entanglement entropy
and the minimum R\'enyi entropy for torus links $T_{p,pn}$ can be interpreted
in terms of the volume of the moduli space of flat connections on certain
Riemann surfaces. We also analyze the R\'enyi entropies of $T_{p,pn}$ link in
the double scaling limit of $k \to \infty$ and $n \to \infty$ and propose that
the entropies converge in the double limit as well.
