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You searched for subject:(Virtual braids). Showing records 1 – 3 of 3 total matches.

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University of Toronto

1. Chterental, Oleg. Virtual Braids and Virtual Curve Diagrams.

Degree: PhD, 2015, University of Toronto

There is a well-known injective homomorphism from the n-strand braid group Bn into Aut(Fn), the automorphism group of a free group on n symbols, first described by Artin. This homomorphism induces an action of Bn on Fn that can be recovered by considering the braid group as the mapping class group of Hn (an upper half plane with n punctures) acting naturally on the fundamental group of Hn. Kauman introduced virtual links [Ka] as an extension of the classical notion of a link in R3. There is a corresponding notion of a virtual braid, and the set of virtual braids on n strands forms a group VBn. In this thesis, we will generalize the Artin action to virtual braids. We will define a set, VCDn, of "virtual curve diagrams" and define an action of VBn on VCDn. Then, we will show that, as in Artin's case, the action is faithful. This provides a combinatorial solution to the word problem in VBn. In the papers [B, M], an extension of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that this extension is not injective by exhibiting a non-trivial virtual braid in the kernel when n = 4. Advisors/Committee Members: Bar-Natan, Dror, Mathematics.

Subjects/Keywords: curve diagrams; virtual braids; 0405

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APA (6th Edition):

Chterental, O. (2015). Virtual Braids and Virtual Curve Diagrams. (Doctoral Dissertation). University of Toronto. Retrieved from http://hdl.handle.net/1807/70904

Chicago Manual of Style (16th Edition):

Chterental, Oleg. “Virtual Braids and Virtual Curve Diagrams.” 2015. Doctoral Dissertation, University of Toronto. Accessed August 06, 2020. http://hdl.handle.net/1807/70904.

MLA Handbook (7th Edition):

Chterental, Oleg. “Virtual Braids and Virtual Curve Diagrams.” 2015. Web. 06 Aug 2020.

Vancouver:

Chterental O. Virtual Braids and Virtual Curve Diagrams. [Internet] [Doctoral dissertation]. University of Toronto; 2015. [cited 2020 Aug 06]. Available from: http://hdl.handle.net/1807/70904.

Council of Science Editors:

Chterental O. Virtual Braids and Virtual Curve Diagrams. [Doctoral Dissertation]. University of Toronto; 2015. Available from: http://hdl.handle.net/1807/70904

2. Cisneros de la Cruz, Bruno Aarón. Caractérisation topologique de tresses virtuelles : Topological characterization of virtual braids.

Degree: Docteur es, Mathématiques, 2015, Université de Bourgogne

Le but de cette thèse est de fournir une caractérisation topologique de tresses virtuelles. Les tresses virtuelles sont des classes d’équivalence de diagrammes de type tresses tracés sur le plan. La relation d’équivalence est générée par l’isotopie, les mouvements de Reidemeister et les mouvements de Reidemeister virtuels. L’ensemble des tresses virtuelles est munie d’une opération de groupe. On parlera alors du groupe de tresses virtuelles. Dans le Chapitre 1, nous introduisons les notions de base de la théorie de noeuds virtuels, nous évoquons certains propriétés du groupe tresses virtuelles, et des liens qu’il a avec le groupe de tresses classiques. Dans le Chapitre 2, nous introduisons la notion de diagramme de Gauss tressé (ou diagramme de Gauss horizontal), et on démontre qu’il s’agit là d’une bonne réinterprétation combinatoire pour les tresses virtuelles. On généralise en particulier certains résultats connus en théorie de noeuds virtuels. Un application est de retrouver la présentation classique du groupe de tresses virtuelles pures à l’aide des diagrammes de Gauss tressés. Dans le Chapitre 3, on introduit les tresses abstraites et on montre qu’elles sont en correspondance bijective avec les tresses virtuelles. Les tresses abstraites sont des classes d’équivalence des diagrammes de type tresses tracés sur une surface orientable avec deux composantes de bord. La relation d’équivalence est générée par l’isotopie, la compatibilité, la stabilité et les mouvements de Reidemeister. La compatibilité est la relation d’équivalence générée par les difféomorphismes préservant l’orientation. La stabilité est la relation d’équivalence générée par l’addition ou la suppression d’anses à la surface, dans le complémentaire du diagramme. Dans le Chapitre 4, on démontre que tout tresse abstraite admets une unique représentant de genre minimal, à compatibilité et mouvements de Reidemeister prés. En particulier, les tresses classiques se plongent dans les tresses abstraites.

The purpose of this thesis is to give a topological characterization of virtual braids. Virtual braids are equivalence classes of planar braid-like diagrams identified up to isotopy, Reidemeister and virtual Reidemeister moves. The set of virtual braids admits a group structure and is called the virtual braid group. In Chapter 1 we present a general introduction to the theory of virtual knots, and we discuss some properties of virtual braids and their relations with classical braids. In Chapter 2 we introduce braid-Gauss dia- grams, and we prove that they are a good combinatorial reinterpretation of virtual braids. In particular this generalizes some results known in virtual knot theory. As an application, we use braid-Gauss diagrams to recover a well known presentation of the pure virtual braid group. In Chapter 3 we introduce abstract braids and we prove that they are in a bijective cor- respondence with virtual braids. Abstract braids are equivalence classes of braid-like diagrams on an orientable surface with two boundary components. The equivalence…

Advisors/Committee Members: Paris, Luis (thesis director).

Subjects/Keywords: Noeuds virtuels; Tresses virtuelles; Théorie de noeuds; Théorie de groupes; Virtual knots; Virtual braids; Knot theory; Group theory; 515

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Cisneros de la Cruz, B. A. (2015). Caractérisation topologique de tresses virtuelles : Topological characterization of virtual braids. (Doctoral Dissertation). Université de Bourgogne. Retrieved from http://www.theses.fr/2015DIJOS025

Chicago Manual of Style (16th Edition):

Cisneros de la Cruz, Bruno Aarón. “Caractérisation topologique de tresses virtuelles : Topological characterization of virtual braids.” 2015. Doctoral Dissertation, Université de Bourgogne. Accessed August 06, 2020. http://www.theses.fr/2015DIJOS025.

MLA Handbook (7th Edition):

Cisneros de la Cruz, Bruno Aarón. “Caractérisation topologique de tresses virtuelles : Topological characterization of virtual braids.” 2015. Web. 06 Aug 2020.

Vancouver:

Cisneros de la Cruz BA. Caractérisation topologique de tresses virtuelles : Topological characterization of virtual braids. [Internet] [Doctoral dissertation]. Université de Bourgogne; 2015. [cited 2020 Aug 06]. Available from: http://www.theses.fr/2015DIJOS025.

Council of Science Editors:

Cisneros de la Cruz BA. Caractérisation topologique de tresses virtuelles : Topological characterization of virtual braids. [Doctoral Dissertation]. Université de Bourgogne; 2015. Available from: http://www.theses.fr/2015DIJOS025

3. Lee, Peter. The Pure Virtual Braid Group is Quadratic.

Degree: 2012, University of Toronto

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this thesis we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.

PhD

Advisors/Committee Members: Bar-Natan, Dror, Mathematics.

Subjects/Keywords: Pure Virtual Braids; Quadraticity; Graded 1-Formal; Universal Finite Type Invariant; 0405

…them going forward. Thus, as mentioned earlier, pure virtual braids are given by the… …of ‘m-singular virtual braids’ – essentially virtual to interpret the ideals IK braids… …such semi-virtual braids, particularly the syzygy known as the Zamolodchikov tetrahedron:4 4… …Chapter 1. Executive Summary 2 The group of pure flat braids PfBn is given by the same… …quadraticity of the pure braid group. The proof relied on the geometry of braids embedded in R3 . In… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Lee, P. (2012). The Pure Virtual Braid Group is Quadratic. (Doctoral Dissertation). University of Toronto. Retrieved from http://hdl.handle.net/1807/32806

Chicago Manual of Style (16th Edition):

Lee, Peter. “The Pure Virtual Braid Group is Quadratic.” 2012. Doctoral Dissertation, University of Toronto. Accessed August 06, 2020. http://hdl.handle.net/1807/32806.

MLA Handbook (7th Edition):

Lee, Peter. “The Pure Virtual Braid Group is Quadratic.” 2012. Web. 06 Aug 2020.

Vancouver:

Lee P. The Pure Virtual Braid Group is Quadratic. [Internet] [Doctoral dissertation]. University of Toronto; 2012. [cited 2020 Aug 06]. Available from: http://hdl.handle.net/1807/32806.

Council of Science Editors:

Lee P. The Pure Virtual Braid Group is Quadratic. [Doctoral Dissertation]. University of Toronto; 2012. Available from: http://hdl.handle.net/1807/32806

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