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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.

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.

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.

Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.

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Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in Homeo+(S1)

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in Homeo+(S1)