Summary: 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 \(\mathrm{SU}(2)_k\) Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of \(k \rightarrow \infty\). We present a detailed analysis of 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 Rényi 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 Rényi 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 Rényi 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 Rényi entropies of \(T_{ p,pn }\) link in the double scaling limit of \(k \rightarrow \infty\) and \(n \rightarrow \infty\) and propose that the entropies converge in the double limit as well.