Статті в журналах: "Braid group"

1

Nasution, Mahyuddin K. M. "The braid group: redefining." MATEC Web of Conferences 197 (2018): 01005. http://dx.doi.org/10.1051/matecconf/201819701005.

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The role of the braid group constitutes one of the invariant measurements. Through the classification of braids formed several parts of the braid group, but does not computationally distinguish them. Some characteristics have been expressed to give the features to a braid in braid group based on redefining relation between braid group and permutation group.

2

Paul, Kamakhya, Pinkimani Goswami, and Madan Mohan Singh. "ALGEBRAIC BRAID GROUP PUBLIC KEY CRYPTOGRAPHY." jnanabha 52, no. 02 (2022): 218–23. http://dx.doi.org/10.58250/jnanabha.2022.52225.

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The braid group cryptography arises with the involvement of the braid group, which is an infinite non-commutative group arising from geometric braids. In this paper, we have proposed a new public key cryptosystem based on braid group. The security of our proposed scheme is based on two hard problems on braid group, conjugacy search problem and p-th root problem on braid group. We also checked the one-wayness, semantic security and efficiency of our proposed scheme, and found it to be computationally secured

3

El-Ghoul, M., and M. M. Al-Shamiri. "Retraction of Braid and Braid Group." Asian Journal of Algebra 1, no. 1 (December 15, 2007): 1–9. http://dx.doi.org/10.3923/aja.2008.1.9.

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4

El-Ghoul, M., and M. M. Al-Shamiri. "Retraction of Braid and Braid Group*." Asian Journal of Algebra 3, no. 1 (December 15, 2009): 8–16. http://dx.doi.org/10.3923/aja.2010.8.16.

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5

Bardakov, Valeriy G., Slavik Jablan, and Hang Wang. "Monoid and group of pseudo braids." Journal of Knot Theory and Its Ramifications 25, no. 09 (August 2016): 1641002. http://dx.doi.org/10.1142/s0218216516410029.

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6

Rodr�guez-Romo, Suemi. "Braid group symmetries." International Journal of Theoretical Physics 30, no. 11 (November 1991): 1403–8. http://dx.doi.org/10.1007/bf00675607.

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7

KAUFFMAN, LOUIS H., and SOFIA LAMBROPOULOU. "A CATEGORICAL MODEL FOR THE VIRTUAL BRAID GROUP." Journal of Knot Theory and Its Ramifications 21, no. 13 (October 24, 2012): 1240008. http://dx.doi.org/10.1142/s0218216512400081.

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This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

8

Häring-Oldenburg, Reinhard. "Braid lift representations of Artin's Braid Group." Journal of Knot Theory and Its Ramifications 09, no. 08 (December 2000): 1005–9. http://dx.doi.org/10.1142/s0218216500000591.

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We recast the braid-lift representation of Contantinescu, Lüdde and Toppan in the language of B-type braid theory. Composing with finite dimensional representations of these braid groups we obtain various sequences of finite dimensional multi-parameter representations.

9

Chang, Wonjun, Byung Chun Kim, and Yongjin Song. "An infinite family of braid group representations in automorphism groups of free groups." Journal of Knot Theory and Its Ramifications 29, no. 10 (August 5, 2020): 2042007. http://dx.doi.org/10.1142/s0218216520420079.

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The [Formula: see text]-fold ([Formula: see text]) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator [Formula: see text] of braid group inside mapping class group induced by [Formula: see text]-fold covering is the product of [Formula: see text] Dehn twists on the surface.

10

Fedoseev, Denis A., Vassily O. Manturov, and Zhiyun Cheng. "On marked braid groups." Journal of Knot Theory and Its Ramifications 24, no. 13 (November 2015): 1541005. http://dx.doi.org/10.1142/s0218216515410059.

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In this paper, we introduce [Formula: see text]-braids and, more generally, [Formula: see text]-braids for an arbitrary group [Formula: see text]. They form a natural group-theoretic counterpart of [Formula: see text]-knots, see [V. O. Manturov; Reidemeister moves and groups, preprint (2014), arXiv:1412.8691]. The underlying idea used in the construction of these objects — decoration of crossings with some additional information — generalizes an important notion of parity introduced by the second author (see [V. O. Manturov, Parity in knot theory, Sb. Math. 201(5) (2010) 693–733]) to different combinatorically geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.

11

Fenn, Roger, Richárd Rimányi, and Colin Rourke. "The braid-permutation group." Topology 36, no. 1 (January 1997): 123–35. http://dx.doi.org/10.1016/0040-9383(95)00072-0.

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12

Constantinescu, Florin. "Unitary braid group representations." Journal of Mathematical Physics 36, no. 6 (June 1995): 3126–33. http://dx.doi.org/10.1063/1.531357.

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13

MOULTON, VINCENT. "HOPF-BRAID GROUPS." Journal of Knot Theory and Its Ramifications 07, no. 08 (December 1998): 1107–17. http://dx.doi.org/10.1142/s0218216598000619.

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In this note we define the Hopf-braid group, a group that is directly related to the group of motions of n mutually distinct lines through the origin in [Formula: see text], which is better known as the braid group of the two-sphere. It is also related to the motion group of the Hopf link in the three-sphere. This relationship is provided by considering the link of a union of complex lines through the origin in [Formula: see text] (i.e. the intersection of the lines with the unit 3-sphere centered at the origin in [Formula: see text]). Through the study of this group we also illustrate some of the connections between the field of knots and braids and that of hyperplane arrangements.

14

WU, JIE. "A BRAIDED SIMPLICIAL GROUP." Proceedings of the London Mathematical Society 84, no. 3 (April 29, 2002): 645–62. http://dx.doi.org/10.1112/s0024611502013370.

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By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.

15

Manturov, Vassily Olegovich, and Igor Mikhailovich Nikonov. "On braids and groups Gnk." Journal of Knot Theory and Its Ramifications 24, no. 13 (November 2015): 1541009. http://dx.doi.org/10.1142/s0218216515410096.

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In [Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208.] the first author gave the definition of [Formula: see text]-free braid groups [Formula: see text]. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to [Formula: see text]-free braid groups for important cases [Formula: see text]. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

16

Szepietowski, B. "Embedding the braid group in mapping class groups." Publicacions Matemàtiques 54 (July 1, 2010): 359–68. http://dx.doi.org/10.5565/publmat_54210_04.

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17

Humphries, Stephen P. "Finite Hurwitz braid group actions for Artin groups." Israel Journal of Mathematics 143, no. 1 (December 2004): 189–222. http://dx.doi.org/10.1007/bf02803499.

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18

KÁDÁR, ZOLTÁN, PAUL MARTIN, ERIC ROWELL, and ZHENGHAN WANG. "LOCAL REPRESENTATIONS OF THE LOOP BRAID GROUP." Glasgow Mathematical Journal 59, no. 2 (June 10, 2016): 359–78. http://dx.doi.org/10.1017/s0017089516000215.

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AbstractWe study representations of the loop braid groupLBnfrom the perspective of extending representations of the braid group$\mathcal{B}$n. We also pursue a generalization of the braid/Hecke/Temperlely–Lieb paradigm – uniform finite dimensional quotient algebras of the loop braid group algebras.

19

Laver, Richard. "Braid group actions on left distributive structures, and well orderings in the braid groups." Journal of Pure and Applied Algebra 108, no. 1 (April 1996): 81–98. http://dx.doi.org/10.1016/0022-4049(95)00147-6.

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20

Brendle, Tara E., and Dan Margalit. "The level four braid group." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 735 (February 1, 2018): 249–64. http://dx.doi.org/10.1515/crelle-2015-0032.

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AbstractBy evaluating the Burau representation att=-1, one obtains a symplectic representation of the braid group. We study the resulting congruence subgroups of the braid group, namely, the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup is equal to the group generated by squares of Dehn twists. We also show that the image of the Brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup.

21

FENN, ROGER, EBRU KEYMAN, and COLIN ROURKE. "THE SINGULAR BRAID MONOID EMBEDS IN A GROUP." Journal of Knot Theory and Its Ramifications 07, no. 07 (November 1998): 881–92. http://dx.doi.org/10.1142/s0218216598000462.

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22

Nasution, M. K. M. "Fuzzy braid group: A concept." Journal of Physics: Conference Series 1116 (December 2018): 022032. http://dx.doi.org/10.1088/1742-6596/1116/2/022032.

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23

Jacak, J., I. Jóźwiak, and L. Jacak. "Composite Fermions in Braid Group Terms." Open Systems & Information Dynamics 17, no. 01 (March 2010): 53–71. http://dx.doi.org/10.1142/s1230161210000059.

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A new implementation of composite fermions, and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group, in accordance with a cyclotron motion of 2D charged particle systems. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with braid subgroup structure. It is demonstrated that composite fermions and composite anyons are rightful 2D particles (not an auxiliary construction) associated with cyclotron braid subgroups instead of the full braid group, which may open a new opportunity for non-Abelian composite anyons for quantum information processing applications.

24

Brandenbursky, Michael, and Jarek Kędra. "Concordance group and stable commutator length in braid groups." Algebraic & Geometric Topology 15, no. 5 (November 12, 2015): 2861–86. http://dx.doi.org/10.2140/agt.2015.15.2861.

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25

FRANKO, JENNIFER M., ERIC C. ROWELL, and ZHENGHAN WANG. "EXTRASPECIAL 2-GROUPS AND IMAGES OF BRAID GROUP REPRESENTATIONS." Journal of Knot Theory and Its Ramifications 15, no. 04 (April 2006): 413–27. http://dx.doi.org/10.1142/s0218216506004580.

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We investigate a family of (reducible) representations of the braid groups [Formula: see text] corresponding to a specific solution to the Yang–Baxter equation. The images of [Formula: see text] under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of [Formula: see text] factoring over Temperley–Lieb algebras and the corresponding link invariants.

26

Droms, Carl, Jacques Lewin, and Herman Servatius. "Tree groups and the 4-string pure braid group." Journal of Pure and Applied Algebra 70, no. 3 (March 1991): 251–61. http://dx.doi.org/10.1016/0022-4049(91)90072-a.

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27

Cai, Shuya, and Hao Li. "Equivariant Mapping Class Group and Orbit Braid Group." Chinese Annals of Mathematics, Series B 43, no. 4 (July 2022): 485–98. http://dx.doi.org/10.1007/s11401-022-0341-6.

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28

Abdulrahim, Mohammad N., and Madline Al-Tahan. "Krammer's Representation of the Pure Braid Group,." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–10. http://dx.doi.org/10.1155/2010/806502.

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We consider Krammer's representation of the pure braid group on three strings: , where and are indeterminates. As it was done in the case of the braid group, , we specialize the indeterminates and to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammer's representation of the pure braid group, .

29

Calvez, Matthieu, and Tetsuya Ito. "A Garside-theoretic analysis of the Burau representations." Journal of Knot Theory and Its Ramifications 26, no. 07 (March 21, 2017): 1750040. http://dx.doi.org/10.1142/s0218216517500407.

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We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of simply-nested braids.

30

Bardakov, Valeriy G., and Jie Wu. "On virtual cabling and a structure of 4-strand virtual pure braid group." Journal of Knot Theory and Its Ramifications 29, no. 10 (August 17, 2020): 2042002. http://dx.doi.org/10.1142/s021821652042002x.

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This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [Formula: see text]. We define simplicial group [Formula: see text] and its simplicial subgroup [Formula: see text] which is generated by [Formula: see text]. Consequently, we describe [Formula: see text] as HNN-extension and find presentation of [Formula: see text] and [Formula: see text]. As an application to classical braids, we find a new presentation of the Artin pure braid group [Formula: see text] in terms of the cabled generators.

31

Zhu, Jun. "On Singular Braids." Journal of Knot Theory and Its Ramifications 06, no. 03 (June 1997): 427–40. http://dx.doi.org/10.1142/s0218216597000285.

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In Vassiliev theory, there is a natural monoid homomorphism from n-strand singular braids to the group algebra of n-strand braid group. J. Birman conjectured that this monoid homomorphism is injective. We show that the monoid homomorphism is injective on braids with up to three singularities and that Birman's conjecture is equivalent to that singular braids are distinguishable by Vassiliev braid invariants.

32

MARGALIT, DAN, and JON McCAMMOND. "GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP." Journal of Knot Theory and Its Ramifications 18, no. 01 (January 2009): 1–20. http://dx.doi.org/10.1142/s0218216509006859.

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We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.

33

JORDAN-SANTANA, MERCEDES. "A GEOMETRIC PROOF THAT THE SINGULAR BRAID GROUP IS TORSION FREE." Journal of Knot Theory and Its Ramifications 16, no. 08 (October 2007): 1067–82. http://dx.doi.org/10.1142/s0218216507005634.

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34

KAMADA, S. "AN OBSERVATION OF SURFACE BRAIDS VIA CHART DESCRIPTION." Journal of Knot Theory and Its Ramifications 05, no. 04 (August 1996): 517–29. http://dx.doi.org/10.1142/s0218216596000308.

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A surface braid is a generalization of classical braids, which is related to classical and 2-dimensional knot theory. It is described by a diagram on a 2-disk called a chart. We prove that surface braids are in one-to-one correspondence to such diagrams modulo some elementary moves. It helps us to handle surface braids. As an application we calculate the Grothendieck group of the semi-group of surface braids. A theorem on symmetric equivalence for the braid group is also given.

35

East, James. "THE FACTORIZABLE BRAID MONOID." Proceedings of the Edinburgh Mathematical Society 49, no. 3 (October 2006): 609–36. http://dx.doi.org/10.1017/s0013091504001452.

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AbstractIn this paper we study the factorizable braid monoid (also known as the merge-and-part braid monoid) introduced by Easdown, East and FitzGerald in 2004. We find several presentations of this monoid, and uncover an interesting connection with the singular braid monoid. This leads to the definition of the flexible singular braid monoid, which consists of ‘flexible-vertex-isotopy’ classes of singular braids. We conclude by defining and studying the pure factorizable braid monoid, the maximal subgroups of which are (isomorphic to) quotients of the pure braid group.

36

Fu, Zhi Yu, Lu Bin Hang, Hai Xu, Jin Cai, Huai Qiang Bian та Jiu Ru Lu. "Rotations of 4nπ Research and no Twisted Mechanism Example Analysis Based on the Theory of Braid Group". Applied Mechanics and Materials 536-537 (квітень 2014): 1355–60. http://dx.doi.org/10.4028/www.scientific.net/amm.536-537.1355.

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The Cable or Pipe between the Two Relatively Rotating Platforms Exists the Twisted Problem. from Viewpoint of Braid Group Theory, Rope’s Twisted State is Researched on. Based on the Characteristics of a Special Garside Braids Δn and Δnk , the Equivalence of Two Rotation Modes is Revealed. also, that the Determination of Rotation Mode’s Minimum Rotate Range is 4∏ while Using Braid Theory is Proposed. the Theory of Braid Group can Not only be Used as a Criterion for Determine Whether the Cable is Twisted, Finally, but also can be Used as Avoid Cable Twisted during Mechanism Design.

37

Bode, Benjamin. "Real algebraic links in S3 and braid group actions on the set of n-adic integers." Journal of Knot Theory and Its Ramifications 29, no. 06 (May 2020): 2050039. http://dx.doi.org/10.1142/s021821652050039x.

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We construct an infinite tower of covering spaces over the configuration space of [Formula: see text] distinct nonzero points in the complex plane. This results in an action of the braid group [Formula: see text] on the set of [Formula: see text]-adic integers [Formula: see text] for all natural numbers [Formula: see text]. 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 [Formula: see text] an infinite sequence of braids, whose braid types are invariants of [Formula: see text]. We present computations for the cases of [Formula: see text] and [Formula: see text] and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials [Formula: see text].

38

BELLINGERI, P., and A. CATTABRIGA. "HILDEN BRAID GROUPS." Journal of Knot Theory and Its Ramifications 21, no. 03 (March 2012): 1250029. http://dx.doi.org/10.1142/s0218216511009534.

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Let H g be a genus g handlebody and MCG 2n( T g) be the group of the isotopy classes of orientation preserving homeomorphisms of T g = ∂ H g, fixing a given set of 2n points. In this paper we study two particular subgroups of MCG 2n( T g) which generalize Hilden groups defined by Hilden in [Generators for two groups related to the braid groups, Pacific J. Math.59 (1975) 475–486]. As well as Hilden groups are related to plat closures of braids, these generalizations are related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.

39

Jeong, Chan-Seok, and Yong-Jin Song. "SPLITTINGS FOR THE BRAID-PERMUTATION GROUP." Journal of the Korean Mathematical Society 40, no. 2 (March 1, 2003): 179–93. http://dx.doi.org/10.4134/jkms.2003.40.2.179.

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40

Nasution, Mahyuddin K. M. "A computation in the braid group." IOP Conference Series: Materials Science and Engineering 725 (January 21, 2020): 012101. http://dx.doi.org/10.1088/1757-899x/725/1/012101.

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41

Allocca, Michael P., Steven T. Dougherty, and Jennifer F. Vasquez. "Signed permutations and the braid group." Rocky Mountain Journal of Mathematics 47, no. 2 (April 2017): 391–402. http://dx.doi.org/10.1216/rmj-2017-47-2-391.

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42

Krammer, Daan. "The braid group of ℤ n". Journal of the London Mathematical Society 76, № 2 (28 вересня 2007): 293–312. http://dx.doi.org/10.1112/jlms/jdm049.

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43

Erlijman, Juliana. "New subfactors from braid group representations." Transactions of the American Mathematical Society 350, no. 1 (1998): 185–211. http://dx.doi.org/10.1090/s0002-9947-98-02007-8.

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44

Todorov, I. T., and L. K. Hadjiivanov. "Monodromy representations of the braid group." Physics of Atomic Nuclei 64, no. 12 (December 2001): 2059–68. http://dx.doi.org/10.1134/1.1432899.

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45

Formanek, Edward. "Braid Group Representations of Low Degree." Proceedings of the London Mathematical Society s3-73, no. 2 (September 1996): 279–322. http://dx.doi.org/10.1112/plms/s3-73.2.279.

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46

Akimenkov, A. M. "Subgroups of the braid group B4." Mathematical Notes of the Academy of Sciences of the USSR 50, no. 6 (December 1991): 1211–18. http://dx.doi.org/10.1007/bf01158260.

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47

Krammer, Daan. "The braid group B4 is linear." Inventiones mathematicae 142, no. 3 (December 2000): 451–86. http://dx.doi.org/10.1007/s002220000088.

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48

Rowell, Eric C., and Zhenghan Wang. "Localization of Unitary Braid Group Representations." Communications in Mathematical Physics 311, no. 3 (November 26, 2011): 595–615. http://dx.doi.org/10.1007/s00220-011-1386-7.

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49

Clark, Sean, and David Hill. "Quantum Supergroups V. Braid Group Action." Communications in Mathematical Physics 344, no. 1 (April 19, 2016): 25–65. http://dx.doi.org/10.1007/s00220-016-2630-y.

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50

Baldwin, John A., and J. Elisenda Grigsby. "Categorified invariants and the braid group." Proceedings of the American Mathematical Society 143, no. 7 (February 26, 2015): 2801–14. http://dx.doi.org/10.1090/s0002-9939-2015-12482-3.

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