Abstract: We present a framework for exploring topologically unique quadrangulations of an input shape. First, the input shape is segmented into surface patches. Second, different topologies are enumerated and explored in each patch. This is realized by an efficient subdivision-based quadrangulation algorithm that can exhaustively enumerate all mesh topologies within a patch. To help users navigate the potentially huge collection of variations, we propose tools to preview and arrange the results. Furthermore, the requirement that all patches need to be jointly quadrangulatable is formulated as a linear integer program. Finally, we apply the framework to shape-space exploration, remeshing, and design to underline the importance of topology exploration