Annular Khovanov-Lee homology, braids, and cobordisms

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a (Z circle plus Z) filtered complex whose filtered chain homotopy type is an invariant of the isotopy class of L subset of (A x I). Using ideas of Ozsvath-Stipsicz-Szabo [31] as reinterpreted by Livingston [30], we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness.