  Rhoades defined a skein action of the symmetric group on noncrossing set
partitions 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 skein
relations that generalize the Ptolemy relation. The skein action on noncrossing
set partitions has seen applications to coinvariant theory and coordinate rings
of partial flag varieties. In this paper, we will show how Rhoades'
$\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module
generated by matchings, thereby explaining how Rhoades' generalized skein
relations all arise from the Ptolemy relation.
