  We study the multi-party entanglement structure of states in Chern-Simons
theory created by performing the path integral on 3-manifolds with linked torus
boundaries, called link complements. For gauge group $SU(2)$, the wavefunctions
of these states (in a particular basis) are the colored Jones polynomials of
the corresponding links. We first review the case of $U(1)$ Chern-Simons theory
where these are stabilizer states, a fact we use to re-derive an explicit
formula for the entanglement entropy across a general link bipartition. We then
present the following results for $SU(2)$ Chern-Simons theory: (i) The
entanglement entropy for a bipartition of a link gives a lower bound on the
genus of surfaces in the ambient $S^3$ separating the two sublinks. (ii) All
torus links (namely, links which can be drawn on the surface of a torus) have a
GHZ-like entanglement structure -- i.e., partial traces leave a separable
state. By contrast, through explicit computation, we test in many examples that
hyperbolic links (namely, links whose complements admit hyperbolic structures)
have W-like entanglement -- i.e., partial traces leave a non-separable state.
(iii) Finally, we consider hyperbolic links in the complexified $SL(2,C)$
Chern-Simons theory, which is closely related to 3d Einstein gravity with a
negative cosmological constant. In the limit of small Newton constant, we
discuss how the entanglement structure is controlled by the Neumann-Zagier
potential on the moduli space of hyperbolic structures on the link complement.
