Questions tagged [kirby-calculus]
Kirby diagrams of 4-manifolds, Kirby moves and Kirby calculus, Akbulut diagrams, handle decompositions
33 questions
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Kirby calculus and local moves
Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
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Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)
A celebrated theorem of Rohlin states the following
An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero.
Simple homological arguments ...
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Construction of invariants of 4-manifolds with the Kirby calculus
I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
...
13
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2
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Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?
It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which ...
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Cobordism and Kirby calculus
It may be a simple question but I wonder to ask:
Is it possible to draw a homology cobordism between $3$-manifolds by using the techniques of Kirby calculus?
At least, for instance, Brieskorn ...
user150450
10
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3
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Doubles of 2-handlebodies
Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. ...
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10
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Kirby calculus on Mazur manifolds
I have questions about Akbulut and Kirby's paper Mazur manifolds.
I couldn't figure out the following equality passages:
Any help will be appreciated.
user150450
9
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2
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A knot in the solid torus and a Mazur manifold
Part 1: The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130:
He indicates that the knot drawn in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \...
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2
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Presentations of exotic 4-manifolds
TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed).
Are there known presentations of $4$-manifolds $M$ with exotic structures, ...
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8
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Inverse Kirby knot
Given an (oriented framed) knot $K$ in the 3-sphere $S^3$, we can perform a surgery along $K$ to get another 3-manifold $M$. From $M$, we can perform the inverse surgery back to $S^3$.
However, the ...
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7
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Simple question on Kirby move
From hyperbolic volume computation, I found that the following two 3-manifolds are (possibly orientation-reversal) homeomorphic:
surgery on figure-eight knot $4_1$, with slope $-5$, and
surgery on $...
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Akbulut's cork involution
Akbulut's cork is the Mazur manifold $W$ shown in the picture below,
This manifold carries an involution of it's boundary $f:\partial W\to \partial W$ that exchanges a meridian of the 0-framed curve ...
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6
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Framings for 2-surgeries on 4-manifolds
I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$.
Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible ...
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Kirby diagram of Enriques surface (as the "(1/2) K3 surface")
Not to be confused with $E(1)\cong\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, which is also known as a $\frac{1}{2}K3$ surface (in the sense that removing a neighbourhood of a regular torus fiber in $E(...
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Kirby diagrams of Mazur manifolds
In the 1980's, Fintushel-Stern and Fickle independently proved that Brieskorn spheres $\Sigma(2,3,25)$ and $\Sigma(3,5,19)$ bound some Mazur type contractible 4-manifolds with a single $0$-, $1$, and $...
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Kirby diagrams: sliding 1-handles over 1-handles and ribbon disks
Consider the Kirby diagram $ D$ given by a 2-component unlink, both dotted circles.
In general, when performing a 1-handle slide over another 1-handle, the band chosen must not link any dotted circle,...
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2
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Are there Kirby diagrams with 3-handles?
Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\...
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Kirby diagram of the complement of a subhandlebody of a smooth closed 4-manifold
Let $X$ be a smooth closed connected 4-manifold. It admits a handlebody structure, having a unique 0- and a unique 4-handle. We can express the handlebody structure as a Kirby diagram (https://en....
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Normal form of framed links under Kirby moves
It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a ...
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Finding an overtwisted disk in a contact surgery diagram
I've been reading through Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci & Stipsicz and have been stuck on an exercise. Consider the following contact surgery diagram for $(S^3,\xi_{-...
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Kirby's theorem for 4-manifolds
In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related ...
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4
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Blow-up and Blow-down kirby local moves for non-orientable $3$-manifold
Can anyone explain or give a reference about the Blow-up and Blow-down Kirby local moves for non-orientable $3$-manifolds?
Thanks, advance.
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Cap product for (co)homology from handle decompositions/Kirby diagrams
Since handle decompositions and Morse functions are intimately related, I'm imagining that a given explicit handle decomposition allows for an explicit description of the cellular complex and thus of (...
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3
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1
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A Mazur manifold bounded by $\Sigma(2,3,13)$
Using Kirby calculus, Akbulut and Kirby first analyzed that the Brieskorn sphere $\Sigma(2,3,13)$ is diffeomorphic to the following link in their famous paper:
Then they switched the circles when ...
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Obtaining the bounding 4-manifold from the Heegaard diagram
It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...
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Picturing twisting of strands explicitly after blow downs
In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
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What's a completely computational/syntactical model for handle decompositions of manifolds?
Simplicial sets, CW complexes
Simplicial sets can be described completely algebraically, by specifying a family of sets, and maps between them satisfying certain relations. This description can be ...
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Lefschetz Fibrations and disk bundles
When reading chaptes 7 of Akbulut's book about $4-$manifolds, he describes a handle decomposition for a manifold given a Lefschetz fibration over $S^2$. The idea is to extend the preimage of a disk ...
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0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots
A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...
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Homotopy type of a 3-manifold produced via Dehn surgery?
My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.
I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
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The Kirby diagram of a manifold glued along the lens space $L(p,1)$
Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
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Tubular neighbourhoods are unique up to ambient isotopy?
Let $M$ be a closed smooth submanifold of $N$. It is well known that tubular neigbourhoods of $M$ are diffeomorphic to the normal bundle of $M$ in $N$ and therefore to each other. Are they smoothly ...
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Mapping class group and surgery theory
Given a smooth manifold $M$ of dimension $n$ and a diffeomorphism $\phi: M \to M$, we can construct a smooth cobordism of dimension $(n+1)$ from $M$ to $M$ by gluing $M \times [0,1]$ with itself by $\...
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