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2
votes
0answers
163 views

What invariants are of great concern in the field of 3-manifolds and why? How much do we know about them? [closed]

I am curious about 3-manifolds though I know little. Here I am trying to know what invariants people in this field are interested in. The following are what I have known and what I particularly want ...
6
votes
1answer
225 views

Is there “nonorientable Heegaard Floer homology”?

I have a Heegaard diagram which produces a non-orientable 3-manifold. I want to know any 3-manifold invariant which can be calculated from Heegaard diagrams for non-orientable 3-manifold. (As far as I ...
4
votes
2answers
192 views

Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed ...
2
votes
1answer
149 views

Looking for software that computes intersection numbers (Heegaard Diagrams)

As a part of my research I am working with intersection matrices of Heegaard diagrams. Is there some software that could help me compute such matrices for some examples? Thanks.
9
votes
1answer
282 views

Heegaard genus of the hyperbolic dodecahedral space (is it 3 or 4?)

I have a question on the hyperbolic dodecahedral space, first described by C.Weber and H. Seifert in 1933 [Die beiden Dodekaederr\"aume, Math Z. 37 (1933), 237-253]. Is it known whether it admits a ...
11
votes
2answers
295 views

The homeomorphism problem for hyperbolic 3-manifolds and the virtual Haken theorem

If $N$ and $N'$ are two closed hyperbolic 3-manifolds, then one would like to have an algorithm which determines whether or not $N$ and $N'$ are homeomorphic. If $N$ and $N'$ are Haken, then such an ...
1
vote
0answers
171 views

Toral decomposition

I have a couple of questions on the following theorem: Theorem. (Jaco, Shalen) Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection ...
10
votes
2answers
489 views

Vector field on 3-sphere

Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...
3
votes
0answers
118 views

Euclidean realisation of a polyhedral complex

Let us say that an Euclidean polyhedral manifold is a manifold that is glued from a finite number of Euclidean polyhedrons by identifying isometrically their co-dimension $1$ faces. Let us assume that ...
6
votes
4answers
236 views

Does Dehn filling always decrease Gromov norm?

As an immediate consequence of Proposition 6.5.2 of Thurston's notes, we have that, if $M$ is a compact 3-manifold with toric boundary and $\tilde M$ is obtained from $M$ via Dehn filling, then for ...
-1
votes
1answer
214 views

mapping class group of a surface

I want to know what techniques are known to present a diffeomorphism on a surface with boundary (the diffeomorphism is not necessarily the identity restricted to the boundary) as product of Dehn ...
1
vote
1answer
173 views

isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$? For each free homotopy classes $\gamma$ of mappings of the circle ...
1
vote
2answers
290 views

Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
2
votes
2answers
202 views

question on Thurston-Bennequin number

I have three questions actually: 1- is it true that in a sufficiently small neighborhood of Legendrian knot in a 3-manifold we can find another Legendrian knot? 2- If the above is true, suppose we ...
16
votes
3answers
493 views

Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...
7
votes
2answers
184 views

How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group which can be generated by n meridians, but not less than n, is an n-bridge knot. There is a one-sentence update, saying that ...
1
vote
1answer
137 views

Two links with the same signatures but unknown if they are related by Kirby moves

I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that: the signatures of $L_1, L_2$ are known. we do not know if they are related by Kirby moves. If so, could you specify the ...
3
votes
1answer
303 views

Geometrization & JSJ decomposition with boundary

Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
4
votes
1answer
253 views

Casson invariant and signature

In W. Neumann, J. Wahl, "Casson invariant of links of singularities", Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...
0
votes
1answer
185 views

A question on Cayley graphs and hyperbolic 3-manifolds

There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric. ...
3
votes
2answers
263 views

Unstable Foliations

Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$. Definition. $\mathcal{F}$ is said to be a unstable ...
4
votes
0answers
150 views

Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken ...
1
vote
1answer
193 views

open book decomposition of graph manifolds

Let's define a graph manifold as 3-manifold which is obtained by plumbing of a circle bundle over $\Sigma _g$ (with Euler number 0) with a circle bundle over $\Sigma_h$ (with Euler number k). Is there ...
0
votes
0answers
95 views

Open Book Decompositions of M^3's : Finding the Projection Map (Hope in Coordinates) in an Abstract Open Book

all: I want to know how to find out , hopefully in coordinates, (but I'll take what's available) , the description of the projection map in an abstract open-book decomposition. Open book ...
5
votes
1answer
249 views

Andreev's Theorem and Thurston's hyperbolization theorem

I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, ...
5
votes
2answers
311 views

The fundamental group of a $3$-manifold with a boundary of genus $>0$

Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma_g$, a surface of genus $g>0$. I would like to find a reference to the following two statements. 1) $\pi_1(M)\ne 0$. 2) ...
5
votes
1answer
263 views

Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
2
votes
0answers
116 views

Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry. To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes. ...
2
votes
1answer
81 views

A non-separating surface in a 3-Manifold with Semibundle structure

I have a question of the non-separating surfaces contained in a 3-Manifold with semi-bundle structure. Suppose $M$ is a $3-$manifold with semi-bundle structure, this means that $M$ is a closed ...
5
votes
1answer
444 views

sufficient conditions on Non-Haken manifolds

Is there an algorithm to detect the Non-Haken Manifold? Or, is there a sufficient condition for a manifold to be a non-Haken manifold? (off course, I hope that condition is not the ones in its ...
7
votes
1answer
201 views

Heegaard genus of covers of cusped hyperbolic 3-manifold

Is there a cusped (non-compact) orientable hyperbolic 3-manifold $M$ which has Heegaard genus $g$, and has a finite-sheeted cover with Heegaard genus $<g$? Moreover, if the cover is index $n$, then ...
9
votes
1answer
344 views

What are the homotopy classes of two-component links in $\mathbb{RP}^3$?

This question comes from an unanswered question on Math Stack Exchange. A two-component link in $\mathbb{RP}^3$ is any embedding $S^1\uplus S^1\to \mathbb{RP}^3$. Two such links are homotopic if ...
1
vote
0answers
257 views

contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?
1
vote
1answer
175 views

Contact structures and adjunction inequality in 3-manifolds

It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the ...
0
votes
0answers
82 views

algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $SL(2,C)$-character variety

Does the following statement "Let $G$ be a finitely generated group and let $X(G)$ be the $SL(2,\mathbb{C})$ character variety of $G$. If $X(G)$ contains an irreducible component $X_0$ ...
2
votes
1answer
109 views

A self-homeomorphism of $L_{p,q}$ is isotopic to one which preserves heegaard splitting

Consider the lens space $L_{p,q}$, which we can describe using its standard heegaard splitting, i.e. define $L_{p,q}$ as a quotient of two solid tori, identifying meridians on the boundary of one with ...
4
votes
2answers
601 views

Tight vs. overtwisted contact structure

I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight ...
8
votes
1answer
420 views

How fast does Ricci flow converge on the three-sphere?

Suppose I have a metric $g_0$ on the $\mathbb S^3$, and let $g_t$ be the solution to Ricci flow (with surgery) with initial metric $g_0$. What are some general results which give upper bounds on the ...
7
votes
2answers
310 views

Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ ?

Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ where $\Sigma$ is any closed surface? I know of the Hatcher-Thurston classification of incompressible surfaces in 2-bridge ...
1
vote
1answer
145 views

construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class

How do we construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class. I know that the fiber of the Seifert fibration has to be transverse to the surface fiber of ...
12
votes
1answer
364 views

Distinguishing 3-manifolds by homologies of covers

In a blog post on ldtopology, a recent arxiv posting of Lins-Lins is discussed. The main argument of that paper is difficult to algorithmically distinguish two 3-manifolds and to that end the authors ...
8
votes
1answer
198 views

Isotopy classes on the disk and mapping tori

Is the following true? "The conjugacy classes of two homeomorphisms of the n-times punctured disk have isotopic representatives iff the associated mapping tori are homeomorphic." By conjugacy class ...
1
vote
0answers
99 views

Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
6
votes
2answers
358 views

Do different Dehn fillings produce homeomorphic 3-manifolds ?

Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold. Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a hyperbolic structure. ...
6
votes
1answer
292 views

When are isometry groups of hyperbolic 3-manifolds finite?

If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic ...
6
votes
0answers
302 views

A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture: "... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
2
votes
1answer
157 views

Once punctured torus bundles in snappy/twister

I have been trying to learn about snappy's method for encoding once-punctured torus bundles (http://www.math.uic.edu/t3m/SnapPy/manifold.html#snappy.Manifold). As you can see from the link, they are ...
15
votes
1answer
520 views

Can the SL_2 character variety of a three-manifold be nonreduced?

Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ ...
6
votes
1answer
338 views

integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...
7
votes
2answers
210 views

Non-tame 3-manifolds covered by the Euclidean space

An open 3-manifold is tame if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and ...