Placing a Markov model of evolution on a phylogenetic network gives rise to a statistical model that is particularly amenable to algebraic study by representing it as an algebraic variety. For group-based Markov models, these algebraic varieties are t-varieties with rich combinatorial structure. In this talk, we will present recent results regarding the dimension and defining polynomials of these algebraic varieties. We will then discuss how these properties can be used for establishing identifiability and network inference.