Related papers: Exceptional surgeries in 3-manifolds

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g…

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally…

This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory.…

Let K be a non-trivial knot in the 3-sphere and let Y be the 3-manifold obtained by surgery on K with surgery-coefficient 1. Using tools from gauge theory and symplectic topology, it is shown that the fundamental group of Y admits a…

We present three large families of new examples of plumbed 3-manifolds that bound rational homology 4-balls. These are constructed using two operations, also defined here, that preserve the lack of a lattice embedding obstruction to…

We prove that for any knot $K$, there exists a one-vertex triangulation of the $3$-sphere containing an edge forming $K$. The proof is constructive, and based on fully augmented links. We use our method to produce ``complicated'' simplicial…

We produce a rational homology 3-sphere that does not smoothly bound either a positive or negative definite 4-manifold. Such a 3-manifold necessarily cannot be rational homology cobordant to a Seifert fibered space or any 3-manifold…

In 1970, Hirsch asked what kind of compact invariant sets could be part of a hyperbolic set. Here we obtain that, in case such an invariant set is a 3D manifold, it is a connected sum of tori with handles quotiented by involutions.…

We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows…

We describe an algorithm which has enabled us to give a complete list, without repetitions, of all closed oriented irreducible 3-manifolds of complexity up to 9. More interestingly, we have actually been able to give a "name" to each such…

We present a new approach to hyperbolic plugs, via a construction of bicontact plugs on 3-manifolds with boundary that are surface bundles over the circle. The boundary components are quasi transverse tori, and we prove a gluing theorem…

In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds.…

We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the…

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that…

This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a…

In this paper, we define a new algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery along a hyperbolic knot complement in S^3. We establish a Casson type invariant for these 3-manifolds. In the last section, we…

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots…

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional…