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We introduce and study a new action of the symmetric group $\mathfrak{S}_n$on the vector space spanned by noncrossing partitions of $\{1, 2, \dots, n\}$in which the adjacent transpositions $(i, i+1) \in \mathfrak{S}_n$ act onnoncrossing partitions by means of skein relations. We characterize theisomorphism type of the resulting module and use it to obtain newrepresentation theoretic proofs of cyclic sieving results due toReiner-Stanton-White and Pechenik for the action of rotation on various classesof noncrossing partitions and the action of K-promotion on two-row rectangularincreasing tableaux. Our skein relations generalize the Kauffman bracket (orPtolemy relation) and can be used to resolve any set partition as a linearcombination of noncrossing partitions in a $\mathfrak{S}_n$-equivariant way.

A skein action of the symmetric group on noncrossing partitions