  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.
