The results presented in this thesis pertain to two distinct areas of low-dimensional topology. First, we give a classification of small Seifert fibered surgeries on hyperbolic pretzel knots, as well as a near-classification of small Seifert fibered surgeries on hyperbolic Montesinos knots. Along with recent results of Ichihara-Masai [IM13], these results complete the classification of all exceptional Dehn surgeries on arborescent knots. Second, we exhibit an infinite family of smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. A subfamily of these knots is then used to show that the subgroup of the smooth double concordance group consisting of topologically doubly slice knots is infinitely generated. One corollary of these results is that there exist infinitely many rational homology 3-spheres (with nontrivial first homology) that embed topologically, but not smoothly, into the 4-sphere. / text

Thesis advisor: Robert Meyerhoff / Thurston showed that for all but a finite number of Dehn Surgeries on a cusped hyperbolic 3-manifold, the resulting manifold admits a hyperbolic structure. Global bounds on this number have been set, and gradually improved upon, by a number of Mathematicians until Lackenby and Meyerhoff proved the sharp bound of 10, which is realized by the figure-eight knot exterior. We improve this result by proving a stronger version of Gordon’s conjecture: that excluding the figure-eight knot exterior, cusped hyperbolic 3-manifolds have at most 8 non-hyperbolic Dehn Surgeries. To do so we make use of the work of Gabai et. al. from a forthcoming paper which parameterizes measurements of the cusp, then uses a rigorous computer aided search of the space to classify all hyperbolic 3-manifolds up to a specified cusp size. Their approach hinges on the discreteness of manifold points in the parameter space, an assumption which cannot be made if the manifolds have infinite volume. In this paper we also show that infinite-volume manifolds, which must be Free Bicuspid, can have cusp volume as low as 3.159. As such, these manifolds are a concern for any future expansion of the approach of Gabai et. al. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge. / text

The bitwist construction, originally discovered by Cannon, Floyd, and Parry, gives us a new method for finding face pairing descriptions of 3-manifolds. In this paper, I will describe the construction in a way suitable for a more general audience than the original research papers. Along the way, I will describe Dehn Surgery and a set of moves which allows us to change the framings of a link without changing the topology of the manifold obtained by Dehn Surgery. Once the theory has been developed, I will apply it to find several bitwist representations of the Poincaré Sphere and 3-Torus. Finally, I discuss how one might attempt to find a set of moves that can take one bitwist representation of a manifold to any other bitwist representation of the same manifold. / Master of Science

Berge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1. / text

Thesis advisor: Joshua E. Greene / 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). / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: Joshua E. Greene / We develop and implement obstructions to realizing a 3-manifold all of whose prime summands are lens spaces as Dehn surgery on a knot K in the Poincaré homology sphere, and in the process, we determine the knot Floer homology groups of a knot with such a surgery. We show that such a surgery never results in a 3-manifold with more than three non-trivial summands, and that if the result of surgery has exactly three non-trivial summands, then K is isotopic to a regular Seifert fiber. We furthermore identify the only two knots with half-integer lens space surgeries, and thus complete the classification of knots in the Poincaré homology sphere with non-integer lens space surgeries. We lastly show that a lens space L(p, q) that is realized as integer surgery on a knot K is realized as integer surgery on a Tange knot when p ≥ 2g(K). In order to do so, we build on Greene’s work on changemaker lattices and develop the theory of E8-changemaker lattices. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Since our world is experienced locally in three-dimensional space, students of mathematics struggle to visualize and understand objects which do not fit into three-dimensional space. 3-manifolds are locally three-dimensional, but do not fit into 3-dimensional space and can be very complicated. Twist and bitwist are simple constructions that provide an easy path to both creating and understanding closed, orientable 3-manifolds. By starting with simple face pairings on a 3-ball, a myriad of 3-manifolds can be easily constructed. In fact, all closed, connected, orientable 3-manifolds can be developed in this manner. We call this work a tool kit to emphasize the ease with which 3-manifolds can be developed and understood applying the tools of twist and bitwist construction. We also show how two other methods for developing 3-manifolds–Dehn surgery and Heegaard splitting–are related to the twist and bitwist construction, and how one can transfer from one method to the others. One interesting result is that a simple bitwist construction on a 3-ball produces a group of manifolds called generalized Sieradski manifolds which are shown to be a cyclic branched cover of S^3 over the 2-braid, with the number twists determined by the hemisphere subdivisions. A slight change from bitwist to twist causes the knot to become a generalized figure-eight knot.

Dans cette thèse, on s'intéresse à la construction et à la déformation de structures CR-sphériques sur des variétés de dimension 3. Pour le faire, on étudie en détail l'espace hyperbolique complexe, son groupe d'isométries et des objets géométriques liés à cet espace. On montre un théorème de chirurgie qui permet de construire des structures CR-sphériques sur des chirurgies de Dehn d'une variété à pointe portant une structure CR-sphérique : il s'applique aux structures de Deraux-Falbel sur le complémentaire du noeud de huit et à celles de Schwartz et de Parker-Will sur le complémentaire de l'entrelacs de Whitehead. On définit aussi les variétés de caractères de groupes de type fini pour les formes réelles de SL(n,C) comme des sous-ensembles de la variété des caractères SL(n,C) fixes par des involutions anti-holomorphes. Ces variétés de caractères, dont on étudie en détail l'exemple du groupe Z/3Z*Z/3Z, fournissent des espaces de déformation pour des représentations d'holonomie de structures CR-sphériques. À l'aide de ces espaces de déformations, et des outils liés aux sphères visuelles dans CP^2, on construit une déformation explicite du domaine de Ford construit par Parker et Will et qui donne une uniformisation CR-sphérique sur le complémentaire de l'entrelacs de Whitehead. Cette déformation fournit une infinité d'uniformisations CR-sphériques sur une chirurgie de Dehn particulière de cette variété, et des uniformisations CR-sphériques sur une infinité de chirurgies de Dehn sur le complémentaire de l'entrelacs de Whitehead. / In this thesis, we study the construction and deformation of spherical-CR structures on three dimensional manifolds. In order to do it, we give a detailed description of the complex hyperbolic plane, its group of isometries and some geometric objects attached to this space such as bisectors and extors. We show a surgery theorem which allows to construct spherical-CR on Dehn surgeries of a cusped spherical-CR manifold : this theorem can be applied for the Deraux-Falbel structure on the figure eight knot complement and for Schwartz's and Parker-Will structures on the Whitehead link complement. We also define the character varieties for a real form of SL(n,C) for finitely generated groups as some subsets of the SL(n,C)-character variety invariant under an anti-holomorphic involution. We study in detail the example of the group Z/3Z*Z/3Z. These character varieties give deformation spaces for the holonomy representations of spherical-CR structures. With these deformation spaces and tools related to the visual spheres of a point in CP^2, we construct an explicit deformation of the Ford domain constructed by Parker and Will, which gives a spherical-CR uniformisation of the Whitehead link complement. This deformation provides infinitely many spherical-CR uniformisations of a particular Dehn surgery of the manifold, and spherical-CR unifomisations for infinitely many Dehn surgeries of the Whitehead link complement.

L'homologie instanton-symplectique est un invariant associé à une variété de dimension trois close orientée, qui a été dé?ni par Manolescu et Woodward, et qui correspond conjecturalement à une version symplectique d'une homologie des instantons de Floer. Dans cette thèse nous étudions le comportement de cet invariant sous l'effet d'une somme connexe, d'une chirurgie de Dehn, et d'un cobordisme de dimension quatre. Nous établissons une formule de Künneth pour la somme connexe : si Y et Y' désignent deux variétés closes orientées de dimension trois, l'homologie instanton-symplectique associée à leur somme connexe est isomorphe à la somme directe du produit tensoriel de leurs groupes d'homologie instantonsymplectique respectifs, et de leur produit de torsion (après décalage des degrés). Nous définissons des versions tordues de cette homologie, et prouvons un analogue de la suite exacte de Floer, reliant les groupes associés à une triade de chirurgie. Cette suite exacte nous permet de calculer le rang des groupes associés à des familles de variétés, notamment les revêtements doubles ramifiés d'entrelacs quasi-alternés, des chirurgies entières de grande pente le long de certains noeuds, ainsi que certaines variétés obtenues par plombage de fibrés en disques au-dessus de sphères. Nous définissons enfin des invariants pour des cobordismes de dimension 4 prenant la forme d'applications entre groupes d'homologie instantonsymplectique des bords, et prouvons que deux des morphismes intervenant dans la suite exacte de chirurgie s'interprètent comme de telles applications, associées aux cobordismes d'attachement d'anses. Nous donnons également un critère d'annulation pour de telles applications associées à des éclatements. / Symplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we study the behaviour of this invariant under connected sum, Dehn surgery, and four-dimensional cobordisms. We prove a Künneth-type formula for the connected sum: let Y and Y' be two closed oriented three-manifolds, we show that the symplectic instanton homology of their connected sum is isomorphic to the direct sum of the tensor product of their symplectic instanton homology, and a shift of their torsion product. We define twisted versions of this homology, and then prove an analog of the Floer exact sequence, relating the invariants of a Dehn surgery triad. We use this exact sequence to compute the rank of the groups associated to branched double covers of quasi-alternating links, some plumbings of disc bundles over spheres, and some integral Dehn surgeries along certain knots. We then define invariants for four dimensional cobordisms as maps between the symplectic instanton homology of the two boundaries. We show that among the three morphisms in the surgery exact sequence, two are such maps, associated to the handle-attachment cobordisms. We also give a vanishing criteria for such maps associated to blow-ups.