Invariance of rowmotion for variants of the Tamari lattice

We show that the rowmotion operator from dynamical algebraic combinatorics behaves the same on all alt $\nu$-Tamari lattices for a fixed lattice path $\nu$. We use this invariance of rowmotion to establish cyclic sieving and homomesy results for rational Tamari lattices. We also conjecture that this invariance extends to the more general cross Tamari lattices.