  As an extension of positive and almost positive diagrams and links, we study
two classes of links we call successively almost positive and weakly
successively almost positive links. We prove various properties of polynomial
invariants and signatures of such links, extending previous results or
answering open questions about positive or almost positive links. We discuss
their minimal genus and fibering property and for the latter prove a fibering
extension of Scharlemann-Thompson's theorem (valid for general links).
