Rotation invariant webs for three row flamingo Specht modules

We introduce a new rotation-invariant web basis for a family of Specht modules $S^{(d^3, 1^{n-3d})}$, indexed by normal plabic graphs satisfying a degree condition and resembling $A_2$ webs. We show that the $\mathfrak{S}_n$ action on our basis can be understood combinatorially via a set of skein relations. From this basis, we obtain a cyclic sieving result for a $q$-analog of the hook length formula for $\lambda$. Our construction extends the jellyfish invariants of Fraser, Patrias, Pechenik, and Striker and is closely related to the weblike subgraphs of Lam.