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A classical theorem of braid theory, dating back to Artin's work, says that two closed braids in a solid torus are ambient isotopic if and only if they represent the same conjugacy class of the braid group. This theorem can be reformulated in the framework of link theory without referring to the group structure. A link in a surface bundle over the circle is transversal whenever it covers the circle. In this terminology, Artin's theorem states that in a solid torus trivially fibered over the circle transversal links are ambient isotopic if and only if they are isotopic in the class of transversal links. We generalize this result by proving that (in the piecewise linear category) transversal links in an arbitrary compact orientable -manifold fibered over the circle with a compact fiber are ambient isotopic if and only if they are isotopic in the class of transversal links. [ABSTRACT FROM AUTHOR]

Let n, m ∈ ℕ, n ≥ 3 and n ≥ m ≥ 2. In this paper, we study some properties of the unrestricted virtual singular braid of braid monoid UVSBmn and group UVSGmn. These properties emerged by studying the analogous properties of the unrestricted virtual singular pseudosymmetric braid of braid monoid and group. These mathematical objects are submonoids and subgroups of the virtual singular braid group VSGn. For many of these quotients, we have obtained reduced presentations. [ABSTRACT FROM AUTHOR]

Representations of braid group B n on n ≥ 2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group V B n on n ≥ 2 strands is an extension of the classical braid group B n by the symmetric group S n . In this paper, we consider flat virtual braid groups F V B n on n ≥ 2 strands and construct a family of representations of F V B n by automorphisms of free groups of rank 2 n . It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n = 2 and proved that the kernel contains a free group of rank two for n ≥ 3 . [ABSTRACT FROM AUTHOR]

This study investigated the effects of different braid materials (polyethylene terephthalate, nylon 6, and their hybrid), weaving frequencies of the braids (7, 9, 11, 13 Hz), and concentrations of polyether sulfone solution (15% and 18 wt%) on braid angles, braid thickness, and bursting resistance. The results showed that for all types of single and hybrid braids, the braid angle decreases with increasing spinning frequency. When spinning frequency was increased from 7 to 13 Hz, the average wall thickness of polyethylene terephthalate, nylon 6, and the hybrid braids increased by approximately 50%, 65%, and 60%, respectively. The bursting pressure results were obtained in the range of 9.3 to 20.2 bars. The Fourier transform infrared spectroscopy results revealed no chemical bond between the braids and the polyether sulfone layer. Optical and electron microscope analyses revealed formation of uniform thickness of the polyether sulfone layers on the surface of the braids, as well as the penetration of polyether sulfone into the gaps between the filaments. The analyses also showed macroscopic adhesion of polyether sulfone to the braids and provided information on the amount and size of the pores in the polymer layers. The results showed that the selection of appropriate weaving frequency, braid materials, and polyether sulfone layer concentration are important as controllable parameters for producing polymeric hollow fiber composites with excellent burst pressure resistance. [ABSTRACT FROM AUTHOR]

In this paper, we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, and a tangle. As a consequence, we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators and defining relations for the fundamental group of compliment of a spatial graph in 3-space ℝ 3 . [ABSTRACT FROM AUTHOR]

A transnational feminist paradigm guides this inquiry when braiding narratives, both literally and abstractly, as a feminist methodology for data analysis. The data include personal narratives (memories and thoughts imprinted on my body, spirit, and mind), master narratives (collaborative databases and public sites), and counter narratives (articles, books, and videos). The braid offers opportunities to (re)imagine coalitional possibilities through figures (and) that materialize in the tension of weaving together. When foregrounding self-reflection in the context of weaving together, I explore supremacy, positionality, and representation. The assemblage of figures leads me to a colleague turned friend turned collaborator who reminds me to work with productive tension toward love. [ABSTRACT FROM AUTHOR]

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is sl n. The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of G n 3 -like groups. The group G n 3 itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of G n 3 -like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the G n 3 -like group structure. [ABSTRACT FROM AUTHOR]

Herein, a theoretical analysis of auxetic tubular braid (ATB) formed with three types of component yarns having different diameter and modulus is presented. A simple theoretical model is developed for predicting the deformation behavior of this structure under uniaxial extension. With its accuracy first validated by the experimental data, the developed model is further corrected based on the real condition and thus becomes capable to characterize the auxetic behavior of the ATB. During the analysis, it is found that the stiff wrap yarn cannot be fully straightened in real condition and its helical angle decreasing rate is inconsistent during the stretching. Herein, an effective tool for understanding the deformation behavior is provided, which is meaningful for the braid's improved design and applications. [ABSTRACT FROM AUTHOR]

Historic Japanese textiles from over 1000 years ago generally show marked deterioration and only very rare examples show their original forms and much information about textile reproduction has been lost. The replication of textile braids lacks systematic methodology and is still being practiced by only few individual braiding experts. The recreation of original braids as close as possible to original braids is a part of Japan's intangible cultural heritage. The aim of this study is to clarify the decision-making procedure through which the braiding experts can decipher the original braiding structures. As a preliminary study of this project, interviews of a braid researcher and a replicating expert, Makiko Tada were performed regarding her working practices. It is important to clarify the braiding parameters for structural analysis such as the number of transits and the balance of ridges, and it became clear that the orientation of multiple colored threads plays an important role. The expert's replicate works were also analyzed using a text-mining statistical technique to clarify the relationship of braiding parameters. The relationship between each braiding parameter and production method such as loop manipulation and Taka-dai became clear. As a result, the process of deciphering the original braid structure has been compiled in simplified workflows, which could contribute to the standardization and improvement in efficiency of replication of cultural property braids. [ABSTRACT FROM AUTHOR]

This article deals with braid groups of complex reflection groups as introduced in [2]. We present results in two directions. First, we extend the works of Gonçalves, Guaschi, Ocampo [5] and Marin [7] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups: we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group. In the second part, we describe the abelianization of the inverse image of a subgroup of a reflection group in its generalized braid group generalizing results of the first author [1]. In particular, these abelianized groups are not necessarily torsion-free. [ABSTRACT FROM AUTHOR]

Let Γg,1m be the mapping class group of the orientable surface Σg,1m of genus g with one parametrized boundary curve and m permutable punctures; when m = 0 we omit it from the notation. Let βm(Σg,1) be the braid group on m strands of the surface Σg,1. We prove that H*(Γg,1m; Z2) ≅ H*(Γg,1; H*(βm(Σg,1); Z2)). The main ingredient is the computation of H*(βm(Σg,1); Z2) as a symplectic representation of Γg,1. [ABSTRACT FROM AUTHOR]

We construct taut foliations in every closed 3-manifold obtained by r-framed Dehn surgery along a positive 3-braid knot K in S³, where r < 2g(K) - 1 and g(K) denotes the Seifert genus of K. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot P(-2, 3, 7), and indeed along every pretzel knot P(-2, 3, q), for q a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by r -framed Dehn surgery along a positive 1-bridge braid in S³, where r < g(K). This paper relies extensively on color figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures. [ABSTRACT FROM AUTHOR]

In this paper, the main idea is to present the solvability of the word problem of pure virtual braid groups by using Gröbner-Shirshov basis theory. [ABSTRACT FROM AUTHOR]

An inversion triple of an element w of a simply laced Coxeter group W is a set { α , β , α + β } , where each element is a positive root sent negative by w. We say that an inversion triple of w is contractible if there is a root sequence for w in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions of elements of W, and have been used to define or characterize certain classes of elements of W, e.g., the fully commutative elements and the freely braided elements. Also, the study of inversion triples is connected with the representation theory of affine Hecke algebras and double affine Hecke algebras. In this paper, we describe the inversion triples that are contractible, and we give a new, simple characterization of the groups W with the property that all inversion triples are contractible. We also study the natural action of W on the set of all triples of (not necessarily positive) roots of the form { α , β , α + β } . This enables us to prove rather quickly that every triple of positive roots { α , β , α + β } is contractible for some w in W and, moreover, when W is finite, w may be taken to be the longest element of W. At the end of the paper, we pose a problem concerning the aforementioned action. [ABSTRACT FROM AUTHOR]

We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the K(π, 1) conjecture holds for the associated family of Artin groups, this establishes homological stability for these groups. In particular, this recovers and extends Arnol'd's proof of stability for the Artin groups of type A, B and D. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to introduce an innovative transversal tubular braid texture and to study the elastic behavior of its 3 D printed structure comparatively to 3 D printed longitudinal tubular braid texture (maypole) to be used as reinforcement. Design/methodology/approach: Regarding the lack of proper machines for the production of the proposed texture, the structure of samples was produced as a tubular lattice braid texture using a 3 D printer with the fused deposition modeling method subsequent to simulation by Rhinoceros software. The produced specimens were composited by polyurethane resin. The composite samples were evaluated by the split disk mechanical test to obtain their hoop stress. The structures of the reinforced composites were theoretically analyzed by ANSYS software. Findings: The results of the mechanical test and theoretical analysis showed that the composites reinforced with transversal tubular lattice braid have higher strength compared to the composites reinforced with longitudinal ones. This assured that the composite reinforced by transversal tubular lattice braid is reliable to be used as high-performance tube for different applications. Originality/value: Further work is carried out to produce the innovated complex structure continuously by a specially designed machine and fibrous materials to reinforce tubular composites in an industrial continual process to be applied for high-pressure fluids flows. [ABSTRACT FROM AUTHOR]

We consider a family of representations of the braid groups Bn corresponding to a specific solution to the Yang-Baxter equation. The images of the pure braid group Pn, a normal subgroup of Bn, under these representations are extraspecial 2-groups and the images of the braid group Bn are extensions of extraspecial 2-groups. We determine conditions under which any representation of the extraspecial 2-group, Em-1, is faithful. We then show that the irreducible representations of Em-1, constructed by Franko, Rowell and Wang, are faithful if and only if m = 2k or m = 2k - 1 (k odd); where as it is not faithful if m = 2k - 1 (k even). [ABSTRACT FROM AUTHOR]

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi-Yau category is simply connected in the sequel [King and Qiu, Invent. Math., to appear]. [ABSTRACT FROM AUTHOR]

We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids. [ABSTRACT FROM AUTHOR]

In 1994, Moss Sweedler's dog proposed a cryptosystem, known as Barkee's Cryptosystem, and the related cryptanalysis. Its explicit aim was to dispel the proposal of using the urban legend that "Gröbner bases are hard to compute", in order to devise a public key cryptography scheme. Therefore he claimed that "no scheme using Gröbner bases will ever work". Later, further variations of Barkee's Cryptosystem were proposed on the basis of another urban legend, related to the infiniteness (and consequent uncomputability) of non-commutative Gröbner bases; unfortunately Pritchard's algorithm for computing (finite) non-commutative Gröbner bases was already available at that time and was sufficient to crash the system proposed by Ackermann and Kreuzer. The proposal by Rai, where the private key is a principal ideal and the public key is a bunch of polynomials within this principal ideal, is surely immune to Pritchard's attack but not to Davenport's factorization algorithm. It was recently adapted specializing and extending Stickel's Diffie–Hellman protocols in the setting of Ore extension. We here propose a further generalization and show that such protocols can be broken simply via polynomial division and Buchberger reduction. [ABSTRACT FROM AUTHOR]

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras. [ABSTRACT FROM AUTHOR]

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and only if GS(D) contains no edges of weight one. Here GS(D), called the Seifert graph of D, is an edge weighted simple graph obtained from D by identifying each Seifert circle of D as a vertex of GS(D) such that two vertices in GS(D) are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in D and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well-known MFW inequality, which states that the a-span of the HOMFLY polynomial of L is a lower bound of 2b(L)−2, as well as the result of Yamada relating the minimum number of Seifert circles over all link diagrams of L to b(L). [ABSTRACT FROM AUTHOR]

The loop braid group is the motion group of unknotted oriented circles in R 3 . In this paper, we study their representations through the approach inspired by two dimensional topological phases of matter. In principle, the motion of loops in R 3 reduces to the motions of points in a two dimensional sliced plane. We realize this physical picture in terms of braided tensor categories and their braid group representations. [ABSTRACT FROM AUTHOR]

We construct an infinite tower of covering spaces over the configuration space of n − 1 distinct nonzero points in the complex plane. This results in an action of the braid group 𝔹 n on the set of n -adic integers ℤ n for all natural numbers n ≥ 2. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid B an infinite sequence of braids, whose braid types are invariants of B. We present computations for the cases of n = 2 and n = 3 and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials ℝ 4 → ℝ 2 . [ABSTRACT FROM AUTHOR]

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group U q sl (2) at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin–Turaev ribbon type functor for tangles with quandle representations. This theory applies to the unrestricted quantum group U q sl (2) and produces an invariant of links with a gauge class of quandle representations. [ABSTRACT FROM AUTHOR]

A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories 𝒞 with the fusion rules of SO (8) 2 we determine the finite group G for which Rep (D ω G) is braided equivalent to 𝒵 (𝒞). In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point. [ABSTRACT FROM AUTHOR]

The necklace braid group NB n is the motion group of the n + 1 component necklace link L n in Euclidean R 3 . Here L n consists of n pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group NB n , especially those obtained as extensions of representations of the braid group B n and the loop braid group LB n . We show that any irreducible B n representation extends to NB n in a standard way. We also find some non-standard extensions of several well-known B n -representations such as the Burau and LKB representations. Moreover, we prove that any local representation of B n (i.e., coming from a braided vector space) can be extended to NB n , in contrast to the situation with LB n . We also discuss some directions for future study from categorical and physical perspectives. [ABSTRACT FROM AUTHOR]

For an unoriented link 𝒦 , let L (𝒦) be the ropelength of 𝒦. It is known that in general L (𝒦) is at least of the order O ((Cr (𝒦)) 3 / 4) , and at most of the order O (Cr (𝒦) ln 5 (Cr (𝒦)) where Cr (𝒦) is the minimum crossing number of 𝒦. Furthermore, it is known that there exist families of (infinitely many) links with the property L (𝒦) = O (Cr (𝒦)). A long standing open conjecture states that if 𝒦 is alternating, then L (𝒦) is at least of the order O (Cr (𝒦)). In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant a > 0 such that L (𝒦) ≥ a B (𝒦) for any 𝒦 , where B (𝒦) is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of 𝒦 (called the maximum braid index of 𝒦). Consequently, L (𝒦) ≥ O (Cr (𝒦)) for any link 𝒦 whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links. [ABSTRACT FROM AUTHOR]

El artículo analiza el arte de la cestería, el cordaje y la fabricación de muebles tradicionales seguido por los artesanos indígenas de México. El artículo también analiza el proceso de trenzado, preparación de fibras y teñido en cestería.

The branched virtual fibering theorem by Sakuma states that every closed orientable 3-manifold with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus g surface bundles as branched double covers of the 3-sphere behaves like 1/ g. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed 3-manifolds are branched double covers of the 3-sphere branched over links. A feature of surface bundles coming from our construction is that the monodromies can be read off the braids obtained from the links as the branched set. [ABSTRACT FROM AUTHOR]

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus g with b boundary components and n punctures in the cases when g = 0 {g=0} with b and n arbitrary, and when g = 1 {g=1} and b + n {b+n} is at most 3, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type D 4 {D_{4}}. As a corollary, we conclude that the pure mapping class groups are linear in these cases. [ABSTRACT FROM AUTHOR]

In this paper, we provide a numerical tool to study a material's coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielsen–Thurston decomposition of the dynamical system. One can thus detect and identify clusters of space–time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. Unlike previous methods, the scalable computational complexity of our braid-based approach allows the analysis of large amounts of trajectories. [ABSTRACT FROM AUTHOR]

We prove that, for n ≥ 3, the minimal dimension of a model of the classifying space of the braid group |$B_{n}$|⁠ , and of the pure braid group |$P_{n}$|⁠ , with respect to the family of virtually cyclic groups is n. [ABSTRACT FROM AUTHOR]

We give a complete classification of free and non-free multiplicities on the A 3 braid arrangement. Namely, we show that all free multiplicities on A 3 fall into two families that have been identified by Abe-Terao-Wakefield (2007) and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the mathematical properties of a class of alternating links called polyhedral links which have been used to model DNA polyhedra. The motivation of such studies is to provide guidance and aid in the research of the properties of certain DNA molecules. For example, such studies can provide characterizations of the structural complexity of DNA molecules. In an earlier work, Cheng and Jin studied the mathematical properties of such polyhedral links and were able to determine the braid index of a double crossover polyhedral link with 4 turn. However, the braid index of a double crossover polyhedral link with 4.5 turn remained an unsolved problem to this date, even though the graphs that admit the double crossover polyhedral links with 4.5 turn have been synthesized. In this paper, we provide a complete formulation of the braid index of a double crossover polyhedral link with an arbitrary turn number. Our approach is more general and it allows us to completely determine the braid indices for a much larger class of links. In the case of the double crossover polyhedral links, our formulation of the braid index is a simple formula based on a simpler graph used as a template to build the double crossover polyhedral links. [ABSTRACT FROM AUTHOR]

We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. Examples are shown to arise from 2-category theory and from bialgebras. In order to describe the 2-categorical examples, we take a multicategorical approach. We explain how certain braided skew monoidal structures in the 2-categorical setting give rise to braided monoidal bicategories. For the bialgebraic examples, we show that, for a skew monoidal category arising from a bialgebra, braidings on the skew monoidal category are in bijection with cobraidings (also known as coquasitriangular structures) on the bialgebra. [ABSTRACT FROM AUTHOR]

A point in the (N , q) -torus knot in ℝ 3 goes q times along a vertical circle while this circle rotates N times around the vertical axis. In the Lissajous-toric knot K (N , q , p) , the point goes along a vertical Lissajous curve (parametrized by t ↦ (sin (q t + ϕ) , cos (p t + ψ))) while this curve rotates N times around the vertical axis. Such a knot has a natural braid representation B N , q , p which we investigate here. If gcd (q , p) = 1 , K (N , q , p) is ribbon; if gcd (q , p) = d > 1 , B N , q , p is the d th power of a braid which closes in a ribbon knot. We give an upper bound for the 4 -genus of K (N , q , p) in the spirit of the genus of torus knots; we also give examples of K (N , q , p) 's which are trivial knots. [ABSTRACT FROM AUTHOR]

The group of planar (or flat) pure braids on n strands, also known as the pure twin group, is the fundamental group of the configuration space F n , 3 (ℝ) of n labeled points in ℝ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two. [ABSTRACT FROM AUTHOR]

We introduce an algebra of braids and ties (or bt-algebra) of type B. In analogy to the construction of the bt-algebra of type A, we define this bt–algebra of type B through a framization of the Hecke algebra of type B. We find a basis for it, a faithful tensorial representation, and we prove that it supports a Markov trace, from which we derive invariants of classical links in the solid torus. [ABSTRACT FROM AUTHOR]

We define geodesic normal forms for the general series of complex reflection groups G(e, e, n). This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(e, e, n) over the generating set of the presentation of Corran–Picantin. Using these geodesic normal forms, we construct intervals in G(e, e, n) that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B(e, e, n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures. [ABSTRACT FROM AUTHOR]

We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function-valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that n-braids with fractional Dehn twist coefficient larger than n - 1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that n-times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms. [ABSTRACT FROM AUTHOR]

Obtaining HOMFLY-PT polynomials H R 1 , ... , R l for arbitrary links with l components colored by arbitrary SU(N) representations R 1 , ... , R l is a very complicated problem. For a class of rank r symmetric representations, the [r]-colored HOMFLY-PT polynomial H [ r 1 ] , ... , [ r l ] evaluation becomes simpler, but the general answer lies far beyond our current capabilities. To simplify the situation even more, one can consider links that can be realized as a 3-strand closed braid. Recently (Itoyama et al. in Int J Mod Phys A28:1340009, 2013. arXiv:1209.6304), it was shown that H [ r ] for knots realized by 3-strand braids can be constructed using the quantum Racah coefficients (6j-symbols) of U q (s l 2) , which makes easy not only to evaluate such invariants, but also to construct analytical formulas for H [ r ] of various families of 3-strand knots. In this paper, we generalize this approach to links whose components carry arbitrary symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials H [ r 1 ] , [ r 2 ] for the two-component link L7a3 whose components carry [ r 1 ] and [ r 2 ] colors. Using our results for exclusive Racah matrices, it is possible to calculate symmetric-colored HOMFLY-PT polynomials of links for the so-called one-looped links, which are obtained from arborescent links by adding a loop. This is a huge class of links that contains the entire Rolfsen table, all 3-strand links, all arborescent links, and, for example, all mutant knots with 11 intersections. [ABSTRACT FROM AUTHOR]

Twisting a knot K in S³ along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. We prove that if the ratio of the Seifert genus to the slice genus for knots in a twist family limits to 1, then the winding number of K about c equals either zero or the wrapping number. As a key application, if {Kn} or the mirror twist family {Kn} contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that c is a braid axis of K if and only if both {Kn} and {Kn} each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for {Kn} to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore. [ABSTRACT FROM AUTHOR]

The Helon model identifies standard model quarks and leptons with framed braids composed of three ribbons joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts. The twisting and braiding of ribbons composing braided 3-belts are interchangeable, and any braided 3-belt can be written in a pure twist form with trivial braiding, specified by a vector of three multiples of half integers [a, b, c], a topological invariant. This paper identifies the set of braided 3-belts that can be written in a braid only form in which all twisting is eliminated instead. For these braids, an algorithm to calculate the braid word is determined which allows the braid word of every braided 3-belt to be written in a canonical form. It is furthermore demonstrated that the set of braided 3-belts does not form a group due to a lack of isogeny. [ABSTRACT FROM AUTHOR]

In this paper we present two new bases, B H 2 ′ and ℬ H 2 , for the Kauffman bracket skein module of the handlebody of genus 2 H 2 , KBSM (H 2). We start from the well-known Przytycki-basis of KBSM (H 2) , B H 2 , and using the technique of parting we present elements in B H 2 in open braid form. We define an ordering relation on an augmented set L consisting of monomials of all different "loopings" in H 2 , that contains the sets B H 2 , B H 2 ′ and ℬ H 2 as proper subsets. Using the Kauffman bracket skein relation we relate B H 2 to the sets B H 2 ′ and ℬ H 2 via a lower triangular infinite matrix with invertible elements in the diagonal. The basis B H 2 ′ is an intermediate step in order to reach at elements in ℬ H 2 that have no crossings on the level of braids, and in that sense, ℬ H 2 is a more natural basis of KBSM (H 2). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds M that are obtained from H 2 by surgery, since isotopy moves in M are naturally described by elements in ℬ H 2 . [ABSTRACT FROM AUTHOR]

• We propose a methodology for the hierarchical representation of multimodal images. • The hierarchical representation relies on the novel concept of braids of partitions. • We introduce a braid-based method to conduct multimodal image segmentation. • The segmentation method is based on an energy minimization procedure. • The proposed approach is validated on different multimodal images. Hierarchical data representations are powerful tools to analyze images and have found numerous applications in image processing. When it comes to multimodal images however, the fusion of multiple hierarchies remains an open question. Recently, the concept of braids of partitions has been proposed as a theoretical tool and possible solution to this issue. In this paper, we demonstrate the relevance of the braid structure for the hierarchical representation of multimodal images. We first propose a fully operable procedure to build a braid of partitions from two hierarchical representations. We then derive a framework for multimodal image segmentation, relying on an energetic minimization scheme conducted on the braid structure. The proposed approach is investigated on different multimodal images scenarios, and the obtained results confirm its ability to efficiently handle the multimodal information to produce more accurate segmentation outputs. [ABSTRACT FROM AUTHOR]

Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most C ⋅ L N - 2 {C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are "almost reducible" in multiple ways, and act on the curve graph with small translation distance. [ABSTRACT FROM AUTHOR]

In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl group. Using these results, we show firstly that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought of as an infinite version of Tit's solution to the word problem. [ABSTRACT FROM AUTHOR]

In 1947, in the paper "Theory of Braids," Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk B n , denoted by B ̂ n . In this paper, we extend Goldsmith's answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface B n (M) , denoting it by B ̂ n (M). We then give a presentation of the group B ̂ n (M) and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface M to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature. [ABSTRACT FROM AUTHOR]