The inference of phylogenetic networks, essential for understanding evolutionary relationships involving hybridization and horizontal gene transfer, presents formidable challenges in both theory and practice. While standard phylogenetic methods can infer gene trees from genetic data, these trees only indirectly reflect the species network topology due to horizontal inheritance and incomplete lineage sorting. 

Previous research has shown that certain network topologies and numerical parameters can be identified, but gaps remain in understanding the full topology of level-1 phylogenetic networks under the Network Multispecies Coalescent model. In this talk, we will aim to fill these gaps and address both the identifiability of the full topology of the network and the numerical parameters by investigating the ideals defined by quartet concordance factors for topological semi-directed networks.