Invariants of annular links, cobordisms and transverse links from combinatorial link Floer complex

We define an annular concordance invariant and study its properties. When specialized to braids, this invariant gives bounds on band rank. We introduce a modified chain complex to reformulate the invariant. Then, by focusing on a special case, we give a refinement of the transverse invariant $\widehat{\theta }$. We also study the relationship of this invariant with transverse and braid monodromy properties.