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You searched for subject:(virtual knot theory). Showing records 1 – 3 of 3 total matches.

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1. 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 (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

2. Leung, Louis. Classical Lie Algebra Weight Systems of Arrow Diagrams.

Degree: 2010, University of Toronto

The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.

PhD

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

Subjects/Keywords: knot theory; virtual knots; weight systems; 0405

…part of the knot involved is locally a braid. We say an invariant of virtual knots is of type… …n if it vanishes on all virtual knot diagrams with more than n semi-virtual crossings… …knot diagram we can always express it as a linear combination of virtual knot diagrams with… …systems This section is a review of the notion of finite type invariants of virtual knots and… …weight systems and finite type invariants of oriented virtual knots modulo “braid-like… 

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

APA (6th Edition):

Leung, L. (2010). Classical Lie Algebra Weight Systems of Arrow Diagrams. (Doctoral Dissertation). University of Toronto. Retrieved from http://hdl.handle.net/1807/26366

Chicago Manual of Style (16th Edition):

Leung, Louis. “Classical Lie Algebra Weight Systems of Arrow Diagrams.” 2010. Doctoral Dissertation, University of Toronto. Accessed August 06, 2020. http://hdl.handle.net/1807/26366.

MLA Handbook (7th Edition):

Leung, Louis. “Classical Lie Algebra Weight Systems of Arrow Diagrams.” 2010. Web. 06 Aug 2020.

Vancouver:

Leung L. Classical Lie Algebra Weight Systems of Arrow Diagrams. [Internet] [Doctoral dissertation]. University of Toronto; 2010. [cited 2020 Aug 06]. Available from: http://hdl.handle.net/1807/26366.

Council of Science Editors:

Leung L. Classical Lie Algebra Weight Systems of Arrow Diagrams. [Doctoral Dissertation]. University of Toronto; 2010. Available from: http://hdl.handle.net/1807/26366


University of Illinois – Chicago

3. Schneider, Jonathan. Diagrammatic Theories of 1- and 2- Dimensional Knots.

Degree: 2016, University of Illinois – Chicago

A meta-theory is described whereby any diagrammatic knot theory may be defined by specifying diagrams and moves. This is done explicitly in dimensions 1 and 2, with more abstract indication of how to extend the meta-theory to higher dimensions. Several examples are given in dimensions 1 and 2, with information about how the theories are related. A topological model for each theory is described. Particular focus is placed on virtual knot theory and welded knot theory, building on work by Kauffman, Satoh, and Rourke, with new results about Rourke's model of welded knots. Advisors/Committee Members: Kauffman, Louis H. (advisor), Radford, David (committee member), Takloo-Bighash, Ramin (committee member), Licht, Arthur L. (committee member), Culler, Marc (committee member).

Subjects/Keywords: knot theory; knot diagrams; surface knot theory; 2-knot theory; virtual knots; virtual knot theory; welded knots

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

APA (6th Edition):

Schneider, J. (2016). Diagrammatic Theories of 1- and 2- Dimensional Knots. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/20811

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Schneider, Jonathan. “Diagrammatic Theories of 1- and 2- Dimensional Knots.” 2016. Thesis, University of Illinois – Chicago. Accessed August 06, 2020. http://hdl.handle.net/10027/20811.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Schneider, Jonathan. “Diagrammatic Theories of 1- and 2- Dimensional Knots.” 2016. Web. 06 Aug 2020.

Vancouver:

Schneider J. Diagrammatic Theories of 1- and 2- Dimensional Knots. [Internet] [Thesis]. University of Illinois – Chicago; 2016. [cited 2020 Aug 06]. Available from: http://hdl.handle.net/10027/20811.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Schneider J. Diagrammatic Theories of 1- and 2- Dimensional Knots. [Thesis]. University of Illinois – Chicago; 2016. Available from: http://hdl.handle.net/10027/20811

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.