Analyzing reduced density matrices in SU(2) Chern-Simons theory

We investigate the reduced density matrices obtained for the quantum states in the context of 3d Chern-Simons theory with gauge group SU(2) and Chern-Simons level $k$. We focus on the quantum states associated with the $T_{p,p}$ torus link complements, which is a $p$-party pure quantum state. The reduced density matrices are obtained by taking the $(1|p-1)$ bi-partition of the total system. We show that the characteristic polynomials of these reduced density matrices are monic polynomials with rational coefficients.