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In real world, mutations of genetic sequences are often accompanied by theirrecombinations. Such joint phenomena are modeled by phylogenetic networks.Nakkleh formulated the phylogenetic network reconstruction problem (PNRP) asfollows: Given a family of phylogenetic trees over a common set of taxa, isthere a unique minimal phylogenetic network whose set of spanning treescontains the family? There are different answers to PNRP, since there aredifferent ways to define what a minimal network is (based on differentoptimization criteria). Inspired by ideas from topological data analysis (TDA),we devise lattice-diagram models for the visualization of phylogenetic networksand of filtrations, called the cliquegram and the facegram, respectively, bothgeneralizing the dendrogram model of phylogenetic trees. Both models allow usto solve the PNRP in a rigorous way and free of choosing optimization criteria.The solution to the phylogenetic network and filtration reconstruction processis obtained by taking the join operation of the dendrograms on the lattice ofcliquegrams, and of facegrams, respectively. Furthermore, we show thatcomputing the join-facegram from a given set of dendrograms is polynomial inthe size and number of the input trees. We propose two novel invariants offacegrams, (i) the face-Reeb graph and (ii) the mergegram of a facegram. Weshow the mergegram is 1-Lipschitz stable, while the face-Reeb graph is not. Inparticular, we show that the mergegram is invariant of weak equivalences offiltrations (a stronger form of homotopy equivalence). This new TDA-signature,the mergegram, can be used as a computable proxy for phylogenetic networks andalso, more broadly, for filtrations of datasets, which might be of independentinterest to TDA. To illustrate the utility of those new TDA-tools tophylogenetics, we provide experiments with artificial and benchmark biologicaldata.

Combinatorial Topological Models for Phylogenetic Networks and the Mergegram Invariant