Phylogenetics begins with reconstructing biological family trees from genetic data. Since Nature is not limited to tree-like histories, we use networks to organize our data, and have discovered new polytopes, metric spaces, and simplicial complexes that help us do so. Moreover, we show that the space of phylogenetic trees dually embeds into the Balanced Minimum Evolution polytope, and use this result to find a complex of faces within the subtour-elimination facets of the Symmetric Traveling Salesman polytope, which is shown to be dual to a quotient complex in network space.