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Derive an analytic formula for S_n^{(3)} of the Borromean rings (6_2^3) in SU(2)k Chern–Simons theory

Derive a closed-form analytic expression for the n-th Rényi tri-entropy S_n^{(3)}(A;B;C) of the three-component Borromean rings link 6_2^3 in SU(2)k Chern–Simons theory for general integer n.

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Background

For the Borromean rings in SU(2)k, the authors present the exact wavefunction in terms of q-deformed factorials but do not obtain an analytic formula for the tripartite Rényi multi-entropy.

They provide numerical results for S_2, S_2{(3)}, and GM_2{(3)} as a function of level k, indicating a need for an analytic treatment.

References

For this state, we do not have an analytic formula for $S_n{(3)}$.

— Multi-entropy from Linking in Chern-Simons Theory (2510.18408 - Yuan et al., 21 Oct 2025) in Section 3.3.3 (Borromean rings)