The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relations within and across species. Recombination, reassortment, horizontal gene transfer, and species hybridization constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from tens or hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Ancestral recombination graphs (ARGs) represent potential histories that explicitly accommodate recombination and mutation events across orthologous genomes. However, ARGs are computationally costly to reconstruct and usually become infeasible for more than few tens of genomes. Recently, Topological Data Analysis (TDA) methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. We build on previous TDA developments for detecting and quantifying recombination, and present a novel framework that can be applied to hundreds of genomes and can be interpreted in terms of minimal histories of mutation and recombination events, quantifying the scales and identifying the genomic locations of recombinations. For that aim, we extend the notion of barcodes in persistent homology, largely increasing their sensitivity to recombination, and present a new type of summary graph (topological ARG, or tARG), analogous to ARGs, that capture ensembles of minimal recombination histories. We implement this framework in a software package, called TARGet, and apply it to several examples, including small migration between different populations and horizontal evolution in finches inhabiting the Gal\'apagos Islands.