Rotation invariant webs for three row flamingo Specht modules

Fraser, Patrias, Pechenik, and Striker introduced jellyfish invariants living within the flamingo Specht module S^(dʳ, 1ⁿ⁻ʳᵈ) indexed by ordered set partitions of n with d blocks of size at least r. They show that jellyfish invariants corresponding to noncrossing partitions form a rotationally invariant linearly independent set. We show that a weakening of the noncrossing conditions extends their construction to a basis, though it comes at the cost of rotation invariance. In the case $r=3$, we recover rotation invariance by extending their construction to a certain set of normal plabic graphs we call augmented webs, due to their resemblance to Kuperberg's A₂ webs.