Academic literature on the topic 'Invariants of knots and 3-manifolds'

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.

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.

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.

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

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

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.

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