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Dissertations / Theses on the topic 'Invariants of knots and 3-manifolds'
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1
Tosun, Bulent. "Legendrian and transverse knots and their invariants." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44880.
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In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.
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Cho, Karina Elle. "Enhancing the Quandle Coloring Invariant for Knots and Links." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/hmc_theses/228.
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Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.
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Kuhlmann, Sally Malinda. "Geodesic knots in hyperbolic 3 manifolds." Connect to thesis, 2005. http://repository.unimelb.edu.au/10187/916.
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This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds.<br>Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded.<br>The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially.<br>We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.
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López, Neumann Daniel. "Kuperberg invariants for sutured 3-manifolds." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7036.
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Dans cette thèse, on étudie les invariants quantiques des 3-variétés de Kuperberg, qui sont basées sur les algèbres de Hopf. On montre que, pour les super-algèbres de Hopf involutives, les invariants de Kuperberg s’étendent à la classe, plus générale, des 3-variétés suturées balancées et en particulier aux compléments d’entrelacs. Pour accomplir ceci, on relève plusieurs aspects de la théorie des torsions de Reidemeister au monde des invariants quantiques, tels que la procédure pour tordre des invariants, le calcul de Fox et les structures Spin^c, et on clarifie les aspects de la théorie des algèbres de Hopf auxquels ils correspondent. Quand notre construction est spécialisée au cas d’une algèbre extérieure, on montre qu’elle calcule la torsion de Reidemeister tordue des 3-variétés suturées<br>In this thesis, we study Kuperberg's Hopf algebra approach to quantum invariants of closed 3-manifolds. We show that, for involutive Hopf superalgebras, Kuperberg invariants extend to the more general class of balanced sutured 3-manifolds, and in particular, to link complements. To achieve this, we bring many aspects of Reidemeister torsion theory into the realm of quantum invar-iants, such as twisting, Fox calculus and Spin^c structures and we make clear to which aspects of Hopf algebra theory these correspond. When our construction is specialized to an exterior algebra, we show that it recovers the twisted Reidemeister torsion of sutured 3-manifolds
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Calimici, Giulio. "State sum invariants of combed 3-manifolds." Thesis, Lille 1, 2019. http://www.theses.fr/2019LIL1I018/document.
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Cette thèse concerne la topologie quantique, une branche des mathématiques née dans les années 1980 suite aux travaux de Jones, Drinfeld et Witten. Un exemple fondamental d'invariant quantique des 3-variétés est due à Turaev-Viro en 1992. Leur approche, dans sa forme générale due à Barrett et Westbury, utilise une catégorie de fusion sphérique comme ingrédient principal et consiste en une somme d'états sur un squelette de la 3-variété dont les sommets sont coloriés par les 6j-symboles de la catégorie.Le résultat principal de la thèse est la construction d'un invariant topologique des 3-variétés peignées (c'est-à-dire des 3-variétés munies d'un champ de vecteurs jamais nuls) qui généralise celui de Turaev-Viro. Ce nouvel invariant est défini au moyen d'une catégorie de fusion pivotale et consiste en une somme d'états sur un squelette ramifié représentant la 3-variété peignée.Lorsque la catégorie de fusion pivotale n'est pas sphérique, l'invariant permet en général de distinguer des champs de vecteurs non homotopes sur une même 3-variété. Ceci est montré en considérant une catégorie de fusion pivotale associée à un caractère d'un groupe fini. Pour cette catégorie, l'invariant correspond à l'évaluation par le caractère de la classe d'Euler d'un certain fibré vectoriel de rang 2 associé au champ de vecteurs<br>This thesis concerns quantum topology, a branch of mathematics born in the 1980s after the work of Jones, Drinfeld and Witten. A fundamental example of a quantum invariant of 3-manifolds is due to Turaev-Viro in 1992. Their approach, in its general form due to Barrett and Westbury, uses a spherical fusion category as the main ingredient and consists in a state sum on a skeleton of the 3-manifold whose vertices are colored by the 6j-symbols of the category. The main result of the thesis is the construction of a topological invariant of combed 3-manifolds (that is, of 3-manifolds endowed with a nowhere-zero vector field) which generalizes that of Turaev-Viro. This new invariant is defined by means of a pivotal fusion category and consists in a state sum on a branched skeleton representing the combed 3-manifold. When the pivotal fusion category is not spherical, the invariant allows in general to distinguish non homotopic vector fields on the same 3-manifold. This is proved by considering a pivotal fusion category associated with a character of a finite group. For this category, the invariant corresponds to the evaluation by the character of the Euler class of a certain vector bundle of rank 2 associated to the vector field
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Alsaeed, Suliman. "Local invariants of fronts in 3-manifolds." Thesis, University of Liverpool, 2014. http://livrepository.liverpool.ac.uk/2006756/.
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An invariant is a quantity which remains unchanged under certain classes of transformations. A wave front (or a front) in a 3-manifold is the image of a surface under a Legendrian map. The aim of this thesis is the description of all local invariants of fronts in 3-manifolds. The front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for various orientation and co-orientation settings of the fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, in the case of framed fronts we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts.
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Millichap, Christian R. "Mutations and Geometric Invariants of Hyperbolic 3-Manifolds." Diss., Temple University Libraries, 2015. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/321918.
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Mathematics<br>Ph.D.<br>The main goal of this thesis is to examine the quality of geometric invariants of finite volume hyperbolic 3-manifolds. In particular, we examine how to construct large classes of hyperbolic 3-manifolds that are geometrically similar: they have a number of geometric invariants that are the same, but are non-isometric. Large classes of geometrically similar hyperbolic 3-manifolds provide examples where the minimal geometric data needed to determine M must be quite large. For our constructions, we will use a cut and paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold. As a corollary of this result, we show that the number of hyperbolic knot complements with the same volume and the same initial length spectrum grows at least factorially fast with the volume and the number of twist regions; a similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we show that the knot complements used for this construction are pairwise incommensurable by analyzing their cusp shapes.<br>Temple University--Theses
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Nosaka, Takefumi. "4-fold symmetric quandle invariants of 3-manifolds." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157740.
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Sequin, Matthew James. "Comparing Invariants of 3-Manifolds Derived from Hopf Algebras." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338251228.
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Kroll, Jochen. "A Casson-Lin invariant for knots in homology 3-spheres." [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=967833590.
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Durusoy, Daniel Selahi. "Heegaard Floer homology of certain 3-manifolds and cobordism invariants." Diss., Connect to online resource - MSU authorized users, 2008.
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Rodriguez, Leslie K. "Euler Characteristic of Incompressible Surfaces in 3-Manifolds and Highly-Alternating Knots." Thesis, California State University, Long Beach, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10688597.
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<p> This thesis investigates the intersection between knot theory and the theory of 3-manifolds. 3-manifolds are well-behaved topological spaces that provide a 3-dimensional ambient space in which we study closed loops, also known as knots. Broadly speaking, the results of this thesis relate the topology of the complement of the knot in the ambient 3-manifold to various combinatorial properties of the knot. </p><p> Historically, 3-manifolds have often been studied by analyzing the surfaces they contain. Two classes of surfaces that have been closely connected to the topology and geometry of 3-manifolds are Heegaard surfaces and essential surfaces. The main result of this thesis ties together the existence of essential surfaces in the knot complement in the 3-manifold and the combinatorial properties of the knots themselves with respect to a Heegaard surface of the ambient 3-manifold. In particular, we show that if a knot has a sufficiently complicated alternating diagram with respect to a Heegaard surface, then the knot complement contains no simple essential surfaces. </p><p> In particular we show that the Euler characteristic of an essential surface in the compliment of the knot K is less than or equal to (–1/10)<i> n</i> where <i>K</i> is an <i>n</i>-filling alternating knot diagram.</p><p>
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Wheeler, Russell Clark. "Using symbolic dynamical systems: A search for knot invariants." CSUSB ScholarWorks, 1998. https://scholarworks.lib.csusb.edu/etd-project/3033.
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Hagge, Tobias J. "Graphical calculus for fusion categories and quantum invariants for 3-manifolds." [Bloomington, Ind.] : Indiana University, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3334995.
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Garza, César. "Examples of hyperbolic knots with distance 3 toroidal surgeries in S³." To access this resource online via ProQuest Dissertations and Theses @ UTEP, 2009. http://0-proquest.umi.com.lib.utep.edu/login?COPT=REJTPTU0YmImSU5UPTAmVkVSPTI=&clientId=2515.
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Hepworth, Richard. "Generalized Kreck-Stolz invariants and the topology of certain 3-Sasakian 7-manifolds." Thesis, University of Edinburgh, 2005. http://hdl.handle.net/1842/15011.
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Schirmer, Trenton Frederick. "Two varieties of tunnel number subadditivity." Diss., University of Iowa, 2012. https://ir.uiowa.edu/etd/3379.
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Knot theory and 3-manifold topology are closely intertwined, and few invariants stand so firmly in the intersection of these two subjects as the tunnel number of a knot, denoted t(K). We describe two very general constructions that result in knot and link pairs which are subbaditive with respect to tunnel number under connect sum. Our constructions encompass all previously known examples and introduce many new ones. As an application we describe a class of knots K in the 3-sphere such that, for every manifold M obtained from an integral Dehn filling of E(K), g(E(K))>g(M).
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Tran, Anh Tuan. "The volume conjecture, the aj conjectures and skein modules." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44811.
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This dissertation studies quantum invariants of knots and links, particularly
the colored Jones polynomials, and their relationships with classical invariants like
the hyperbolic volume and the A-polynomial. We consider the volume conjecture that
relates the Kashaev invariant, a specialization of the colored Jones polynomial at a
specific root of unity, and the hyperbolic volume of a link; and the AJ conjecture that
relates the colored Jones polynomial and the A-polynomial of a knot. We establish
the AJ conjecture for some big classes of two-bridge knots and pretzel knots, and
confirm the volume conjecture for some cables of knots.
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Ben, Aribi Fathi. "A study of properties and computation techniques of the L2-Alexander invariant in knot theory." Sorbonne Paris Cité, 2015. http://www.theses.fr/2015USPCC227.
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Cette thèse présente plusieurs propriétés, des valeurs explicites et des techniques de calcul des torsions d'Alexander L2 des variétés de dimension 3 compactes à bord vide ou torique, notamment des extérieurs de noeuds et d'entrelacs. Les torsions d'Alexander L2 sont des généralisations des polynômes d'Alexander tordus, où le complexe de chaînes cellulaires du revêtement universel de la variété est tordu par une représentation du groupe fondamental de la variété sur l'espace des opérateurs d'un espace de Hilbert de dimension infinie. Les torsions d'Alexander L2 des 3-variétés ont été définies en 2014 par J. Dubois, S. Friedl et W. Lück, et généralisent l'invariant d'Alexander L2 des noeuds introduit par W. Li et W Zhang en 2006. Ces torsions sont des invariants topologiques qui sont des classes de fonctions sur les réels strictement positifs. Elles existent uniquement lorsque certaines conditions techniques sont vérifiées, et sont difficiles à calculer en général. Malgré tout, nous pouvons extraire d'importantes informations de ces invariants, comme le volume simplicial de la variété ou la norme de Thurston. Dans cette thèse, nous démontrons que l'invariant d'Alexander L2 des noeuds détecte le noeud trivial. Nous démontrons également une formule de chirurgie de Dehn pour les torsions d'Alexander L2. De même, par diverses techniques, nous calculons explicitement les torsions des extérieurs d'entrelacs toriques dans la sphère S3 et le tore solide, ce qui nous permet de démontrer des formules générales de sommes connexes et de câblages pour les entrelacs<br>This manuscript presents several properties, explicit values and computation techniques of L2-Alexander torsions for compact 3-manifolds with empty or toroidal boundary, especially for knot exteriors and link exteriors. The L 2 -Alexande torsions are generalisations of the twisted Alexander polynomials, as the cellular chain complex of the universal covering of a manifold is twisted by an infinite-dimensional Hilbert representation of the fundamental group of the manifold. The L2-Alexander torsions of 3-manifolds were defined in 2014 by J. Dubois, S. Friedl and W. Lück, and generalize the L2 -Alexander invariant of a knot introduced by W. Li and W. Zhang in 2006. These torsions are topological invariants that are classes of maps on the positive real numbers. They only exist when certain technical conditions are satisfied, and they are hard to compute in general. Despite these difficulties, we are able to extract important information from thes invariants, like the simplicial volume of the manifold or the Thurston norm. In this thesis, we prove that the L2-Alexander invariant of knots detects the trivial knot. We also prove a Dehn surgery formula for the L2-Alexander torsions. Similarly, using various techniques, we compute explicitly the torsions of exteriors of torus links in die 3-sphere and in die solid torus, which leads us to prove general formulas for connected sums and cablings of links
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Krishna, Siddhi. "Taut foliations, positive braids, and the L-space conjecture:." Thesis, Boston College, 2020. http://hdl.handle.net/2345/bc-ir:108731.
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Thesis advisor: Joshua E. Greene<br>We construct taut foliations in every closed 3-manifold obtained by r-framed Dehn surgery along a positive 3-braid knot K in S^3, 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. We adapt our techniques to construct taut foliations in every closed 3-manifold obtained along r-framed Dehn surgery along a positive 1-bridge braid, and indeed, along any positive braid knot, in S^3, where r < g(K)-1. These are the only examples of theorems producing taut foliations in surgeries along hyperbolic knots where the interval of surgery slopes is in terms of g(K)<br>Thesis (PhD) — Boston College, 2020<br>Submitted to: Boston College. Graduate School of Arts and Sciences<br>Discipline: Mathematics
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Meilstrup, Mark H. "Wild Low-Dimensional Topology and Dynamics." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2203.
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In this dissertation we discuss various results for spaces that are wild, i.e. not locally simply connected. We first discuss periodic properties of maps from a given space to itself, similar to Sharkovskii's Theorem for interval maps. We study many non-locally connected spaces and show that some have periodic structure either identical or related to Sharkovskii's result, while others have essentially no restrictions on the periodic structure. We next consider embeddings of solenoids together with their complements in three space. We differentiate solenoid complements via both algebraic and geometric means, and show that every solenoid has an unknotted embedding with Abelian fundamental group, as well as infinitely many inequivalent knotted embeddings with non-Abelian fundamental group. We end by discussing Peano continua, particularly considering subsets where the space is or is not locally simply connected. We present reduced forms for homotopy types of Peano continua, and provide a few applications of these results.
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George, Jennifer Lynn. "TQFTs from Quasi-Hopf Algebras and Group Cocycles." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1369834588.
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Vera, Arboleda Anderson Arley. "Homomorphismes de type Johnson pour les surfaces et invariant perturbatif universel des variétés de dimension trois." Thesis, Strasbourg, 2019. http://www.theses.fr/2019STRAD009/document.
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Soit Σ une surface compacte connexe orientée avec une seule composante du bord. Notons par M le groupe d'homéotopie de Σ. En considérant l'action de M sur le groupe fondamental de Σ, il est possible de définir différentes filtrations de M ainsi que des homomorphismes sur chaque terme de ces filtrations. Le but de cette thèse est double. En premier lieu, nous étudions deux filtrations de M : la " filtration de Johnson-Levine " introduite par Levine et la " filtration de Johnson alternative " introduite recemment par Habiro et Massuyeau. Les définitions de ces deux filtrations prennent en compte un corps en anses bordé par la surface. Nous nous référons à ces filtrations comme " filtrations de type Johnson " et les homomorphismes correspondants sont appelés " homomorphismes de type Johnson " par leur analogie avec la filtration de Johnson originale et les homomorphismes de Johnson usuels. Nous donnons une comparaison de la filtration de Johnson avec la filtration de Johnson-Levine au niveau du monoïde des cobordismes d'homologie de Σ. Nous donnons également une comparaison entre la filtration de Johnson alternative, la filtration Johnson-Levine et la filtration de Johnson au niveau du groupe d'homéotopie. Deuxièmement, nous étudions la relation entre les " homomorphismes de type Johnson" et l'extension fonctorielle de l'invariant perturbatif universel des variétés de dimension trois (l'invariant de Le-Murakami-Ohtsuki ou invariant LMO). Cette extension fonctorielle s'appelle le foncteur LMO et il prend ses valeurs dans une catégorie de diagrammes. Nous démontrons que les "homomorphismes de type Johnson " peuvent être lus dans la réduction arborée du foncteur LMO. En particulier, cela fournit une nouvelle grille de lecture de la réduction arborée du foncteur LMO<br>Let Σ be a compact oriented surface with one boundary component and let M denote the mapping class group of Σ. By considering the action of M on the fundamental group of Σ it is possible to define different filtrations of M together with some homomorphisms on each term of the filtrations. The aim of this thesis is twofold. First, we study two filtrations of M : the « Johnson-Levine filtration » introduced by Levine and « the alternative Johsnon filtration » introduced recently by Habiro and Massuyeau. The definition of both filtrations involve a handlebody bounded by Σ. We refer to these filtrations as ≪ Johnson-type filtrations » and the corresponding homomorphisms have referred to as « Johnson-type homomorphisms » by their analogy with the original Johnson filtration and the usual Johnson homomorphisms. We provide a comparison of the Johnson filtration with the Johnson-Levine filtration at the level of the monoid of homology cobordisms of Σ. We also provide a comparison of the alternative Johnson filtration with the Johnson-Levine filtration and the Johnson filtration at the level of the mapping class group. Secondly, we study the relationship between the « Johnson-type homomorphisms » and the functorial extension of the universal perturbative invariant of 3-manifolds (the Le-Murakami-Ohtsuki invariant or LMO invariant). This functorial extension is calling the LMO functor and it takes values in a category of diagrams. We prove that the « Johnson-type homomorphisms » is in the tree reduction of the LMO functor. In particular, this provides a new reading grid of the tree reduction of the LMO functor
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Gutierrez, Quispe Robert Gerson. "Aspectos de la teoría de nudos." Bachelor's thesis, 2019. http://hdl.handle.net/11086/14649.
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Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2019.<br>Los nudos, tal cual aparecen en nuestra vida cotidiana, son un objeto de estudio en la Matemática. La Teoría de Nudos es la rama de la Matemática que se encarga de su estudio. Un problema central es el de poder decir si dos nudos dados son equivalentes o no. Los matemáticos, en la búsqueda de responder esta pregunta, entre otras, han desarrollado diversas técnicas y herramientas en esta área de estudio. En este trabajo se hace un recorrido en el estudio de la Teoría de Nudos, comenzando con las definiciones más elementales, hasta llegar a estudiar herramientas sofisticadas como el polinomio de Alexander, el grupo de un nudo y las matrices de Seifert, entre otros. En los dos últimos capítulos se investigan los dos temas siguientes: nudos virtuales y presentaciones de Wirtinger. En este último se hace un aporte, dando una nueva familia infinita de presentaciones de Wirtinger no geométricas.<br>The knots we usually see in our lifes are studied in mathematics in the branch called Knot Theory. A main problem is to decide whether two knots are equivalent or not. Many tools and techniques have been developed by mathematicians in order to answer this and other related questions.
In this work, we study Knot Theory from the beginning, with definitions and elementary notions, until some sophisticated concepts and tools like the Alexander polynomial, the knot group and Seifert matrices, among others. In the last two chapters, we work on the following two particular subjects: virtual knots and Wirtinger presentations. In this last one, we made a small contribution by presenting a new infinite family of Wirtinger presentations which are not geometric.<br>Fil: Gutierrez Quispe. Robert Gerson. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
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Zare, Douglas J. "Geometric Invariants in Contact Structures on 3-Manifolds." Thesis, 1999. https://thesis.library.caltech.edu/13593/1/zare-dj-1999.pdf.
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<p>Manifolds that have serious reasons to be odd-dimensional usually carry natural contact structures. — V. I. Arnold</p>
<p>Contact structures are important geometric structures on smooth, odd-dimensional manifolds. This thesis studies contact structures on 3-manifolds, where the theory is enriched by the study of knots and connections to the geometry of 4-manifolds.</p>
<p>In this thesis, we construct a geometric invariant of (framed) knots transverse to contact structures, <i>size</i>, and investigate its connections with Dehn surgeries, symplectic cobordisms, and geometric intersection invariants under contactomorphisms.</p>
<p>In chapter 2, we characterize tight and overtwisted contact structures in terms of the sizes of their transverse knots. We define surgeries of contact 3-manifolds along transverse curves, and give some upperbounds for sizes of knots after such surgeries. We show that certain <i>reducing</i> surgeries are precisely the results of certain symplectic cobordisms. Any transverse knot in a coorientable contact structure has such a family of such surgeries. This suggests that certain contact surgeries preserve tightness, and in section 3.2 we present some partial results in this direction. We provide some methods for obtaining lower bounds for the sizes of knots.</p>
<p>In chapter 3, we study the intersections of knots with surfaces under contactomorphisms. We study the local action of contactomorphisms on arcs near surfaces in contact 3-manifolds and the intersections of overtwisted discs with knots and overtwisted unknots with surfaces.</p>
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Harvey, Shelly Lynn. "Higher-order polynomial invariants of 3-manifolds giving lower bounds for Thurston norm." Thesis, 2002. http://hdl.handle.net/1911/18088.
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We define a new infinite sequence of invariants, d&d1;n for n ≥ 0, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. We show that they give lower bounds for the Thurston norm. Moreover, we show that they give better estimates for the Thurston norm than the previously known bounds given by the Alexander norm, d&d1;0 . To do this, we exhibit 3-manifolds whose Alexander norm is trivial but whose d&d1;n are strictly increasing and can be made arbitrarily large. Other applications are made to detecting 3-manifolds that fiber over S 1 and to detecting 4-manifolds that admit no symplectic structure.
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Kroll, Jochen [Verfasser]. "A Casson-Lin invariant for knots in homology 3-spheres / vorgelegt von Jochen Kroll." 2003. http://d-nb.info/967833590/34.
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