Tied Pseudo Links & Pseudo Knotoids

In this paper, we study the theory of pseudo knots, which are knots with some missing crossing information, and we introduce and study the theory of pseudo tied links and the theory of pseudo knotoids. In particular, we first present a braiding algorithm for pseudo knots and we then introduce the L-moves in that setting, with the use of which we formulate a sharpened version of the analogue of the Markov theorem for pseudo braids. Then, we introduce and study the theory of tied pseudo links, that generalize the notion of tied links, and we exploit the relation between tied pseudo links and tied singular links. We first present an L-move braid equivalence for tied singular braids. Then, we introduce the tied pseudo braid monoid and we formulate and prove analogues of the Alexander and Markov theorems for tied pseudo links. Finally, we introduce and study the theory of pseudo knotoids, that generalize the notion of knotoids. We present an isotopy theorem for pseudo knotoids and we then pass to the level of braidoids. We further introduce and study the pseudo braidoids by introducing the pseudo L-moves and by presenting the analogues of the Alexander and Markov theorems for pseudo knotoids. We also discuss further research related to tied (multi)-knotoids and tied pseudo (multi)-knotoids. The theory of pseudo knots may serve as a strong tool in the study of DNA, while tied links have potential use in other aspects of molecular biology.