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.

作者最近介绍了束缚链和束缚辫状线半身像，并用于定义经典链的新不变量。在这里，我们给出了一个纯粹的代数形式的组合链接的组合形式。借助这个新版本，我们证明了束缚辫状单面体具有类似于半直接群乘积的分解。通过使用这种分解，我们证明了联系链的亚历山大定理和马尔可夫定理。另外，我们介绍了奇异结的奇异结，奇异辫状单体，以及Homflypt型不变式的某些族。这些不变量是五变量多项式。最后，我们研究这些不变量的行为。特别是，我们证明了我们的不变量区分了由Paris–Rabenda不变式无法区分的非同位素奇异链接。