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Rhoades defined a skein action of the symmetric group on noncrossing setpartitions which generalized an action of the symmetric group on matchings. The$\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation,while the action on set partitions is defined in terms of a set of skeinrelations that generalize the Ptolemy relation. The skein action on noncrossingset partitions has seen applications to coinvariant theory and coordinate ringsof partial flag varieties. In this paper, we will show how Rhoades'$\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-modulegenerated by matchings, thereby explaining how Rhoades' generalized skeinrelations all arise from the Ptolemy relation.

An embedding of the skein action on set partitions into the skein action on matchings