  Tied links and the tied braid monoid were introduced recently by the authors
and used to define new invariants for classical links. Here, we give a version
purely algebraic-combinatoric of tied links. With this new version we prove
that the tied braid monoid has a decomposition like a semi--direct group
product. By using this decomposition we reprove the Alexander and Markov
theorem for tied links; also, we introduce the tied singular knots, the tied
singular braid monoid and certain families of Homflypt type invariants for tied
singular links; these invariants are five-variables polynomials. Finally, we
study the behavior of these invariants; in particular, we show that our
invariants distinguish non isotopic singular links indistinguishable by the
Paris-Rabenda invariant.
