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Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called -configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of -configurations in ${ℝ}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of ${\left({ℝ}^{2\times 2}\right)}^4$. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm...