Links of rings offer an intuitive picture of multipartite entanglement. Cutting one ring can free some of the others, just as losing a particle, i.e. tracing it out, can disentangle some of the remaining ones. This analogy classifies multipartite entanglement according to which groups of particles stay entangled when others are lost, a question at the heart of robust quantum networks. However, pure states realizing a given link class were known only in special cases. Here, we close this gap. Recasting link polynomials as hypergraphs, we give an explicit formula for pure qudit states and prove that, for every link class, their entanglement under any pattern of particle loss matches the link exactly. The same construction also controls genuine multipartite entanglement, determining exactly which of the groups that stay entangled are genuinely multipartite entangled, and all of them can be made so. As a consequence, we prove the conjecture that
m-resistant pure qudit states, which stay entangled after the loss of any
m particles but become fully separable after the loss of
m+1, exist for any number of particles and any
m.
We investigate a priori detection probabilities of genuine multipartite entanglement (GME). Even if one does not have knowledge of the basis in which a state is produced by a source, how a channel decoheres it or of the very working of the detectors used, we find that it is possible to detect GME with reasonably high probability in a feasible fashion. We show that by means of certain separability criteria, GME can be detected in a measurement-base-independent way. Our method provides several applications whenever e.g. state tomography is not possible or too demanding, and is a tool to investigate security issues in multi-particle quantum cryptographical protocols.