323,609 research outputs found
Unknotting information from Heegaard Floer homology
No full textWe use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsváth and Szabó's obstruction to unknotting number one. We determine the unknotting numbers of 910, 913, 935, 938, 1053, 10101 and 10120; this completes the table of unknotting numbers for prime knots with crossing number nine or less. Our obstruction uses a refined version of Montesinos' theorem which gives a Dehn surgery description of the branched double cover of a knot
Introduction to Heegaard splittings and Heegaard-Floer Homology.
No full textilustracionesSe estudia a fondo la construcción de las descomposiciones de Heegaard, haciendo énfasis en el por qué es una herramienta central en el estudio de las 3-variedades. Para esto, analizamos las 3-variedades desde las categorías topológica, suave y triángulable, en donde siempre podemos derivar el concepto de descomposición de Heegaard como algo arraigado a la 3-variedad misma. Luego de esto se explora uno de los invariantes de 3-variedades más recientes, la homología de Heegaard-Floer, centrándonos en la aparición de las descomposiciones de Heegaard en su construcción, sirviendo así como eje motivador para futuros proyectos. (Texto tomado de la fuente)The construction of Heegaard decompositions is studied in depth, emphasizing why it is a central tool in the study of 3-manifolds. For this, we analyze the 3-manifolds from the topological, smooth and triangular categories, where we can always derive the Heegaard decomposition concept as something rooted in the 3-manifold itself. After this, one of the most recent 3-manifold invariants is explored, the Heegaard-Floer homology, focusing on the appearance of Heegaard decompositions in its construction, and thus serving as a motivating axis for future projects.MaestríaMagíster en Ciencias - Matemática
The geometry of the disk complex
No full textWe give a distance estimate for the disk complex. We use the distance estimate to prove that the disk complex is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus
Heegaard diagrams and applications
No full textThe main objective of this thesis is to study Heegaard diagrams and their applications.
First, we investigate Heegaard diagrams of closed 3-manifolds and introduce the circle
and chord presentation for a connected, closed 3-manifold. The equivalence problem
for Heegaard diagrams after connected sum moves and Dehn twists will be investigated.
Presentations will be used to detect reducible Heegaard diagrams and homeomorphic
3-manifolds.
We also investigate Heegaard diagrams of the 3-sphere. The main result of this part
is that if two Heegaard diagrams of the 3-sphere have the same genus, then there is a
sequence of connected sum moves and Dehn twists to pass from one to the other. If we
use connected sum moves only, Heegaard curves can be changed to primitive curves and
if we use Dehn twists only Heegaard curves can be brought into a simple position.
Finally, we construct an immersion of a compact, orientable, connected 3-manifold
with non-empty boundary into R³ with at most double and triple points as singularities.
Further, we prove that if the boundary of the 3-manifold consists of 2-spheres and the
3-manifold can immerse into R³ with only double points as singularities, then the 3-
manifold must be a punctured 3-sphere or a punctured (S¹ x S²)# • • • #(S¹ x S²).Science, Faculty ofMathematics, Department ofGraduat
A note on a Heegaard diagram of S^3
No full textOchiai describes a Heegaard diagram of S^3, with a particular associated fundamental group presentation H.This Heegaard diagram has neither waves nor pairsof complementary handles, so it is directly reducible neither by awave-move , nor by a Singer's move of type III'. Hence it is a counterexample to theWhitehead's conjecture and to the algorithm A ofVolodin-Kuznetsov-Fomenko.We construct a crystallization of S^3 having H as an associated presentation of the fundamentalgroup with respect to a suitable choice of generators and relatorsand such that Ochiai's Heegaard diagram is one among the Heegaard diagramsassociated to the crystallization.Moreover, we prove that at least one among the remaining Heegaarddiagrams associated to our crystallization hassome pairs of complementary handles, and so it is directly reducibleby Singer's move of type III' to the canonical diagram ofS^3
Heegaard splittings of graph manifolds
No full textLet M be a totally orientable graph manifold with characteristic submanifold T and
let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is
the amalgamation of strongly irreducible Heegaard splittings. The splitting surfaces S_i of
these strongly irreducible Heegaard splittings have the property that for each vertex
manifold N of M, S_i cap N is either horizontal, pseudohorizontal, vertical or
pseudovertical
Heegaard splittings of branched coverings of 𝑆³
No full textThis paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are “unique” to the question of whether plat representations of knots and links are “unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that
S
3
{S^3}
could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for
π
1
S
3
{\pi _1}{S^3}
are always “nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of
S
3
{S^3}
which are branched over closed 3-braids. It is established that no counterexamples to the “genus 2 Poincaré conjecture” occur in this class of 3-manifolds.</p
Heegaard splittings and virtually special square complexes
No full textWe give a new perspective of Heegaard splittings in terms square complexes
and Guirardel's notion of a \textit{core} which allows for combinatorial
measurement of the obstruction to being a connect sum of Heegaard diagrams. A
Heegaard splitting is a decomposition of a closed orientable -manifold into
two isomorphic handle bodies that have a shared boundary surface. Usually, a
number of curves on the shared boundary surface, called a Heegaard diagram, are
used to describe a Heegaard splitting. We define a larger object, the
\textit{augmented Heegaard diagram}, by building on methods of Stallings and
Guirardel to encode the information of a Heegaard splitting.
\textit{Augmented Heegaard diagrams} have several desirable properties: each
2-cell is a square, they have \textit{non-positive combinatorial curvature} and
they are \textit{virtually special}. Restricting to manifolds that do not have
as a connect summand, augmented Heegaard diagrams are tied to
the decomposition of a -manifold via connect sum as described above.Comment: Revised according to comments received; some misstatements correcte
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The girth of a Heegaard splitting
No full textWe construct simple curves from immersed curves in the setting of handlebodies and
Heegaard splittings. We define a measure of complexity we call girth for closed curves in a
handlebody. We extend this complexity to Heegaard splittings and pose a conjecture about
all Heegaard splittings. We prove a test case of this conjecture. Let S be a compact
surface embedded in the boundary of a handlebody H. Then the minimum girth over all curves
in S can be achieved by a simple closed curve. We also present algorithms to compute the
girth of curves and surfaces
Heegaard splittings of prime 3-manifolds are not unique
No full textThe authors construct an infinite family of prime homology 3-spheres of Heegaard genus 2, satisfying the following two non-uniqueness properties: (1) Each of the manifolds can be structured as the 2-fold cyclic branched cover over each of two inequivalent knots, one of which is a torus knot. (2) Each of the manifolds admits at least two equivalence classes of genus 2 Heegaard splittings. All of the manifolds are Seifert fiber spaces, the properties of which are used to prove (1). The non-uniqueness of Heegaard splittings is based on the work of the first author and H. M. Hilden [Trans. Amer. Math. Soc. 213 (1975), 315–352], who proved that for Heegaard genus 2 splittings of the 2-fold branched cyclic cover of the knot K, the equivalence class of the Heegaard splitting determines uniquely the knot type K. The authors then show that if Σp,q is the 2-fold cyclic branched cover of the torus knot (p,q), then Σp,q is also the 2-fold cyclic branched cover of a knot different from (p,q), and that Σp,q admits a Heegaard splitting of genus 2.NSFAlfred P. Sloan Foundation.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu
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