Aicardi, F.Juyumaya, J.2022-11-302022-11-302021http://repositoriobibliotecas.uv.cl/handle/uvscl/7210Tied 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.© 2021 Elsevier Inc. All rights reserved.TIED LINKSSET PARTITIONBT–ALGEBRAINVARIANTS FOR SINGULAR LINKS AND TIEDSINGULAR LINKSArticulohttps://doi.org/10.1016/j.aim.2021.107629