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In this paper we present recent results on the computation of skein modules of 3-manifolds using braids and appropriate knot algebras. Skein modules generalize knot polynomials in S3 to knot polynomials in arbitrary 3-manifolds and they have become extremely essential algebraic tools in the study of 3-manifolds. In this paper we present the braid approach to the HOMFLYPT and the Kauffman bracket skein modules of the Solid Torus ST and the lens spaces L(p,1) and S1 X S2.

Doubly periodic tangles, or \textit{DP tangles}, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are completely defined by their generating cells, the {\it flat motifs}, which can be chosen in infinitely many ways. DP tangles are used in modelling materials and physical systems of entangled filaments. In this paper we establish the mathematical framework of the topological theory of DP tangles. We first introduce a formal definition of DP tangles as topological objects and proceed with an exhaustive analysis in order to characterize the notion of {\it equivalence} between DP tangles and between their flat motifs. We further generalize our results to other diagrammatic categories, such as framed, virtual, singular, pseudo and bonded DP tangles, which could be used in novel applications.

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.

Tied links in S^3 were introduced by Aicardi and Juyumaya as standard links in S^3 equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces L(p,1), in handlebodies of genus g, and in the complement of the g-component unlink. We introduce the tied braid monoids TMg,n by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an L-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

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.

Tied links in S^3 were introduced by Aicardi and Juyumaya as standard links in S^3 equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces L(p,1), in handlebodies of genus g, and in the complement of the g-component unlink. We introduce the tied braid monoids TMg,n by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an L-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.