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This talk concerns combinatorial models of two symmetric group representations. The first is the fermionic diagonal coinvariant ring, whose top degree piece has a combinatorial model indexed by noncrossing set partitions. The second is a space of $SL_2(\mathbb{C})$ invariant tensors, which has a combinatorial model indexed by noncrossing matchings. We will explore how these representations are connected, and in particular explain how the skein relations used to "resolve crossings" in the set partition model arise from the skein relation in the matching model. Both representations are part of larger families of representations, and we will conclude by examining possible generalizations.