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In this talk we will explore the combinatorial structure of dual equivalence graphs G_lambda. The vertices are Standard Young tableaux of fixed shape lambda that allows us to further understand the combinatorial structure of G_lambda, and the edges are given by dual Knuth equivalences. The graph G_lambda is the 1-skeleton of a cubical complex C_lambda. One can ask whether the cubical complex is CAT(0); this is a desirable metric property that allows us to describe the combinatorial structure of G_lambda very explicitly. We will discuss the CAT(0) characterization of Ardila--Owen--Sullivant. It is constructive and provides an algorithm for determining when a cubical complex is CAT(0). Using their characterization, we prove that C_lambda is CAT(0) if and only if lambda is a hook.