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We study the multi-boundary entanglement structure of the state associatedwith the torus link complement $S^3 \backslash T_{p,q}$ in the set-up ofthree-dimensional SU(2)$_k$ Chern-Simons theory. The focal point of this workis the asymptotic behavior of the R\'enyi entropies, including the entanglemententropy, in the semiclassical limit of $k \to \infty$. We present a detailedanalysis for several torus links and observe that the entropies converge to afinite 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 thesum 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 linkingnumber $n$. Using the analytic techniques, we show that the universal partcomprises of Riemann zeta functions and can be written in terms of thepartition functions of two-dimensional topological Yang-Mills theory. Moreprecisely, it is equal to the R\'enyi entropy of certain states prepared intopological $2d$ Yang-Mills theory with SU(2) gauge group. Further, theuniversal parts appearing in the large $k$ limits of the entanglement entropyand the minimum R\'enyi entropy for torus links $T_{p,pn}$ can be interpretedin terms of the volume of the moduli space of flat connections on certainRiemann surfaces. We also analyze the R\'enyi entropies of $T_{p,pn}$ link inthe double scaling limit of $k \to \infty$ and $n \to \infty$ and propose thatthe entropies converge in the double limit as well.

Semiclassical limit of topological R\'enyi entropy in $3d$ Chern-Simons theory