For a link in a thickened annulus $A \times I$, we define a $\mathbb{ Z} \oplus \mathbb{ Z} \oplus \mathbb{ Z}$ filtration on Sarkar-Seed-Szab\'o's perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.
Categories: math.GT