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Phylogenetic networks are increasingly becoming popular in phylogenetics since they have the ability to describe a wider range of evolutionary events than their tree counterparts. In this talk, we discuss Markov models on phylogenetic networks and their associated algebra, geometry, and combinatorics. A phylogenetic network model is a parameterized collection of discrete probability distributions. Taking the Zariski closure of this set of distributions gives us an algebraic variety and an associated polynomial ideal. Properties of these varieties and ideals can help us establish statistical properties, such as parameter identifiability, a necessary property for precise statistical inference. Assuming the Jukes-Cantor model of DNA evolution and restricting to one reticulation vertex, we will use tools from computational algebraic geometry to show that the semi-directed network topology of large-cycle networks is generically identifiable. This is joint work with Colby Long.