A survey on knotoids, braidoids and their applications
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paper_title: Braids, Links, and Mapping Class Groups.

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The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.
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paper_title: Threading knot diagrams

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Alexander [ 1 ] showed that an oriented link K in S 3 can always be represented as a closed braid. Later Markov [ 5 ] described (without full details) how any two such representations of K are related. In her book [ 3 ], Birman gives an extensive description, with a detailed combinatorial proof of both these results.
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paper_title: Representation of links by braids: A new algorithm

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If a is a braid with n components, the closure of a, denoted #, is constructed by connecting the endpoints at the top level to the bottom endpoints with n standard curves. This procedure yields an oriented link ~ having the same number of crossings as a. A classical result of Alexander [I], [2], [3] states that every oriented link is isotopic to a closed braid #. In his proof Alexander modifies the diagram of an oriented link by a sequence of elementary operations to obtain a closed braid. During this transformation the "geometry of the picture" is completely changed. In many applications of Alexander's algorithms links with few crossings yield closed braid with a large number of crossings. On the other hand many algebraic invariants of links are first defined on braids. If we wish to compute these invariants for a "small" link L, it will be very useful to have the following principle:
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paper_title: Markov’s theorem in 3-manifolds

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Abstract In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in S 3 that sharpens the classical theorem. Then we give a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3-manifold.
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paper_title: A new proof of Markov's braid theorem

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The purpose of this paper is to introduce a new proof of Markov’s braid theorem, in terms of Seifert circles and Reidemeister moves. This means that the proof will be of combinatorial and essentially 2-dimensional nature. One characteristic feature of our approach is that nowhere in the proof will we use or refer to the braid axis. This allows for greater flexibility in various transformations of the diagrams considered. Other proofs of Markov’s theorem can be found in [2], [3], [4] and [5]. As in Vogel’s paper [6], ”closed braid” will be understood to mean a special kind of diagram of an oriented link: the diagram lives on a 2-dimensional sphere and is supposed to have Seifert circles that are nested in one another forming a single chain of concentric circles with compatible orientations. Let us recall that every diagram may be transformed into a closed braid (this is called Alexander’s theorem, see [1]). One particularly nice way to do it is by using reducing moves of Vogel, see [6]. Such a move is simply a type II Reidemeister move that is performed on two arcs belonging to different Seifert circles with orientations as in figure 1. Reduction moves will be the main tool in our proof of Markov’s theorem.
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paper_title: Knots Related by Knotoids

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AbstractTuraev recently introduced the concept of a knotoid as a particular sort of knotted arc. There are several maps from knotoids to ordinary knots, or knotted circles. The two most natural way...
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paper_title: On Markov's Theorem

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We give a new proof of Markov's classical theorem relating any two closed braid representations of the same knot or link. The proof is based upon ideas in a forthcoming paper by the authors,"Stabilization in the braid groups". The new proof of the classical Markov theorem is used by Nancy Wrinkle in her forthcoming manuscript"The Markov Theorem for transverse knots".
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paper_title: A study of braids in 3-manifolds

paper_content:

This work provides the topological background and a preliminary study for the analogue of the 2-variable Jones polynomial as an invariant of oriented links in arbitrary 3- manifolds via normalized traces of appropriate algebras, and it is organized as follows: ::: ::: Chapter 1: Motivated by the study of the Jones polynomial, we produce and present a new algorithm for turning oriented link diagrams in S3 into braids. Using this algorithm we then provide a new, short proof of Markov's theorem and its relative version. ::: ::: Chapter 2: The objective of the first part of Chapter 2 is to state and prove an analogue of Markov's theorem for oriented links in arbitrary 3-manifolds. We do this by modifying first our algorithm, so as to produce an analogue of Alexander's theorem for oriented links in arbitrary 3-manifolds. In the second part we show that the study of links (up to isotopy) in a 3-manifold can be restricted to the study of cosets of the braid groups Bn,m, which are subgroups of the usual braid groups Bn+m . ::: ::: Chapter 3: In this chapter we try to use the above topological set-up in a procedure analogous to the way V.F.R. Jones derived his famous link invariant. The analogy amounts to the following: We observe that Bn,1 - the braid group related to the solid torus and to the lens spaces L(p, 1) - is the Artin group of the Coxeter group of Bn-type. This implies the existence of an epimorphism of eEn,1 onto the Hecke algebra of Bn-type. Then we give an analogue of Ocneanu's trace function for the above algebras. This trace, after being properly normalized, yields a HOMFLY-PTtype isotopy invariant for oriented links inside a solid torus. Finally, by forcing a strong condition, we normalize this trace, so as to obtain a link invariant in SI x S2.
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paper_title: The minimal number of Seifert circles equals the braid index of a link

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A method for the production of textile webs having high porosity and absorbency, with a leather-like surface grain comprising roughening a basic fabric of shrinkable fibers, optionally stretching the fabric if necessary in at least the warp or weft direction; impregnating at least one side of the fabric with an aqueous fluid foam containing a dispersion of a polymeric compound hardenable at a temperature above 120 DEG C, until the foam penetrates at least completely through the pile; drying the impregnated fabric; heating the dried impregnated fabric at 120 DEG to 160 DEG C for a time sufficient to harden said polymeric compound; shrinking the dried impregnated fabric by treatment with hot water or steam; and recovering said textile webs having a high porosity and absorbency with a leather-like surface grain.
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paper_title: Parity in Knotoids

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This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.
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paper_title: New Invariants of Knotoids

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Abstract In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for both classical and virtual knotoids in analogy to corresponding invariants of virtual knots. We show that knotoids in S 2 have symmetric affine index polynomials. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a knotoid in S 2 .
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paper_title: Parity in Knotoids

paper_content:

This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.
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paper_title: A spanning set and potential basis of the mixed Hecke algebra on two fixed strands

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The mixed braid groups \(B_{2,n}, \ n \in \mathbb {N}\), with two fixed strands and n moving ones, are known to be related to the knot theory of certain families of 3-manifolds. In this paper, we define the mixed Hecke algebra \(\mathrm {H}_{2,n}(q)\) as the quotient of the group algebra \({\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}\) over the quadratic relations of the classical Iwahori–Hecke algebra for the braiding generators. We further provide a potential basis \(\Lambda _n\) for \(\mathrm {H}_{2,n}(q)\), which we prove is a spanning set for the \(\mathbb {Z}[q^{\pm 1}]\)-additive structure of this algebra. The sets \(\Lambda _n,\ n \in \mathbb {Z}\) appear to be good candidates for an inductive basis suitable for the construction of Homflypt-type invariants for knots and links in the above 3-manifolds.
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paper_title: New Invariants of Knotoids

paper_content:

Abstract In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for both classical and virtual knotoids in analogy to corresponding invariants of virtual knots. We show that knotoids in S 2 have symmetric affine index polynomials. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a knotoid in S 2 .
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paper_title: Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids

paper_content:

In this paper we introduce a method that offers a detailed overview of the entanglement of an open protein chain. Further, we present a purely topological model for classifying open protein chains by also taking into account any bridge involving the backbone. To this end, we implemented the concepts of planar knotoids and bonded knotoids. We show that the planar knotoids technique provides more refined information regarding the knottedness of a protein when compared to established methods in the literature. Moreover, we demonstrate that our topological model for bonded proteins is robust enough to distinguish all types of lassos in proteins.
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paper_title: Parity in knot theory

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In this work we study knot theories with a parity property for crossings: every crossing is declared to be even or odd according to a certain preassigned rule. If this rule satisfies a set of simple axioms related to the Reidemeister moves, then certain simple invariants solving the minimality problem can be defined, and invariant maps on the set of knots can be constructed. The most important example of a knot theory with parity is the theory of virtual knots. Using the parity property arising from Gauss diagrams we show that even a gross simplification of the theory of virtual knots, namely, the theory of free knots, admits simple and highly nontrivial invariants. This gives a solution to a problem of Turaev, who conjectured that all free knots are trivial. In this work we show that free knots are generally not invertible, and provide invariants which detect the invertibility of free knots. The passage to ordinary virtual knots allows us to strengthen known invariants (such as the Kauffman bracket) using parity considerations. We also discuss other examples of knot theories with parity. Bibliography: 27 items.
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paper_title: New Invariants of Knotoids

paper_content:

Abstract In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for both classical and virtual knotoids in analogy to corresponding invariants of virtual knots. We show that knotoids in S 2 have symmetric affine index polynomials. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a knotoid in S 2 .
---
paper_title: Parity in Knotoids

paper_content:

This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.
---
paper_title: Parity in Knotoids

paper_content:

This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.
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paper_title: Markov’s theorem in 3-manifolds

paper_content:

Abstract In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in S 3 that sharpens the classical theorem. Then we give a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3-manifold.
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paper_title: A study of braids in 3-manifolds

paper_content:

This work provides the topological background and a preliminary study for the analogue of the 2-variable Jones polynomial as an invariant of oriented links in arbitrary 3- manifolds via normalized traces of appropriate algebras, and it is organized as follows: ::: ::: Chapter 1: Motivated by the study of the Jones polynomial, we produce and present a new algorithm for turning oriented link diagrams in S3 into braids. Using this algorithm we then provide a new, short proof of Markov's theorem and its relative version. ::: ::: Chapter 2: The objective of the first part of Chapter 2 is to state and prove an analogue of Markov's theorem for oriented links in arbitrary 3-manifolds. We do this by modifying first our algorithm, so as to produce an analogue of Alexander's theorem for oriented links in arbitrary 3-manifolds. In the second part we show that the study of links (up to isotopy) in a 3-manifold can be restricted to the study of cosets of the braid groups Bn,m, which are subgroups of the usual braid groups Bn+m . ::: ::: Chapter 3: In this chapter we try to use the above topological set-up in a procedure analogous to the way V.F.R. Jones derived his famous link invariant. The analogy amounts to the following: We observe that Bn,1 - the braid group related to the solid torus and to the lens spaces L(p, 1) - is the Artin group of the Coxeter group of Bn-type. This implies the existence of an epimorphism of eEn,1 onto the Hecke algebra of Bn-type. Then we give an analogue of Ocneanu's trace function for the above algebras. This trace, after being properly normalized, yields a HOMFLY-PTtype isotopy invariant for oriented links inside a solid torus. Finally, by forcing a strong condition, we normalize this trace, so as to obtain a link invariant in SI x S2.
---
paper_title: Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids

paper_content:

In this paper we introduce a method that offers a detailed overview of the entanglement of an open protein chain. Further, we present a purely topological model for classifying open protein chains by also taking into account any bridge involving the backbone. To this end, we implemented the concepts of planar knotoids and bonded knotoids. We show that the planar knotoids technique provides more refined information regarding the knottedness of a protein when compared to established methods in the literature. Moreover, we demonstrate that our topological model for bonded proteins is robust enough to distinguish all types of lassos in proteins.
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paper_title: Markov’s theorem in 3-manifolds

paper_content:

Abstract In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in S 3 that sharpens the classical theorem. Then we give a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3-manifold.
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Title: A Survey on Knotoids, Braidoids and Their Applications

Section 1: Knotoids and Knotoid Isotopy
Description 1: Discuss basic concepts and definitions related to knotoids, including knotoid diagrams, isotopy moves, and the distinction between planar and spherical knotoids.

Section 2: Knotoids, Classical Knots, and Virtual Knots
Description 2: Explore the relationship between knotoids and classical knots, the concept of knotoid closures, and how knotoids relate to virtual knots.

Section 3: Geometric Interpretations of Knotoids
Description 3: Present geometric interpretations of both spherical and planar knotoids, including their relation to Θ-graphs and open space curves.

Section 4: Invariants of Knotoids
Description 4: Review existing works on knotoid invariants, such as the bracket polynomial, the Jones polynomial, and novel invariants specific to knotoids.

Section 5: The Theory of Braidoids
Description 5: Introduce fundamental concepts of braidoids, including their diagrams, isotopy moves, and the closure operation.

Section 6: Turning Knotoids into Braidoids
Description 6: Explain the process of converting knotoid diagrams into braidoid diagrams, including the use of L-braidoiding moves and algorithms.

Section 7: Applications of Knotoids and Braidoids
Description 7: Highlight applications of knotoids and braidoids in studying proteins and propose an algebraic encoding method for open protein chains using braidoids.