Questions tagged [kirby-diagram]
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22
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Showing a 4-manifold is contractible
In Gompf-Stipsicz book, we're presented with the Akbulut cork, and a brief explanation of why it is contractible (see below).
Would someone be able to explain what homotopy he is referring too? I ...
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Intersection Form from a Kirby Diagram
If there are only 2 handles and no 1 handle in a Kirby Diagram then the intersection form of the underlying simply-connected 4-manifold coincides with the linking form. But what if there’s at least ...
- 805
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$E(n)\# \mathbb{CP}^2=2n\mathbb{CP}^2\# (10n-1)\overline{\mathbb{CP}^2}$
How does one prove the following statement : $$E(n)\# \mathbb{CP}^2=2n\mathbb{CP}^2\# (10n-1)\overline{\mathbb{CP}^2}$$
Here $E(n)$ denotes the $n$-th elliptic surface formed by fiber summing $E(1)$ $...
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Can framings on plumbed manifolds be taken to be even?
The definition of "plumbed manifold" that I'm using in this context is the following - given a weighted tree $\Gamma$, build up a framed link $L(\Gamma)$ by chaining together two copies of ...
- 12.6k
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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 ...
- 193
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Way to determine the type of a knot given by a diagram
Is there a general way to determine the type of a knot given by a diagram? I am using KLO (Kirby calculator) and I encounter some nontrivial knots while doing this. For example, can we determine the ...
- 2,100
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Are these two links equivalent?
Are the following two links equivalent (orientation preserving isotopies)?
The two links have the same linking number. The only difference is the crossing that in one case is positive while in the ...
- 5,540
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Not every 3-manifold is a graph manifold
Surgery presentations
It is well known that any orientable closed 3-manifold $M$ has a surgery presentation, i.e. can be obtained by doing surgery on $\mathbb{S}^3$ on a link $L$.
We can also ...
- 5,540
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Kirby calculus on E8 plumbing
I was trying to show that the 4-manifold described in Kirby diagram as a E8-plumbing (see the diagram below) has the same boundary as the 2-handlebody on the left-handed trefoil with surgery ...
- 2,153
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Integer homology sphere as subsequent surgery on integer homology spheres
By Lickorish and Wallace , any closed,connected, orientable 3-manifold can be gotten as a surgery on a link in $S^3$. Let say our manifold, M, is an integer homology sphere and L = $ L_1 \cup L_2 \cup ...
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Who invented special framed links as a way to specify four-manifolds?
It is possible to specify (smooth, oriented, compact) four-manifolds with special framed links:
$$I \times \mathbb{R}P^3$$
Such a link is divided in two sublinks, one ordinary link representing the 2-...
- 2,672
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648
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Prerequisites for Kirby Calculus?
I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus?
I have a solid ...
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How can uncountably many closed smooth 4-manifolds be presented by an essentially countable alphabet (Kirby diagrams)?
A smooth, closed 4-manifold admits a handle decomposition which is specified completely by its Kirby diagram. A Kirby diagram, up to isotopy, can be seen as a labelled morphism in the tangle category. ...
- 2,672
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How can a Kirby diagram fail to determine a handle decomposition?
I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
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How to get a Kirby diagram of $S^1 \times M^3$ if $M^3$ is given by a surgery diagram?
In "4-manifolds and Kirby Calculus" by Gompf and Stipsicz, there is a nice description of how to get the Kirby diagram of $S^1 \times M^3$, given a Kirby diagram of $M^3$. Basically, one thickens the ...
- 2,672
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Are there Kirby diagrams for manifolds with boundaries?
There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
- 2,672
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Kirby diagrams for nonorientable $4$-manifolds
In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
- 2,681
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Are there any combinatorial studies of Kirby calculus?
All of the other diagrammatic calculi I know of can be utilised with basically just combinatorial knowledge - for instance calculating knot and link polynomials. Are there similar combinatorial ...
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Cancelling 3-handles in Kirby diagrams
Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 ...
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About Kirby Diagrams
I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 ...
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Kirby-like diagrams for $M^n$ when $n > 4$
Are there any attempts on constructing Kirby-like diagrams for representing manifolds $M^n$ with $n > 4$. What are the references on that ?
I think you run out of dimension in which you can draw ...
- 2,681
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4-Manifolds of which there exist no Kirby diagrams
In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin ...
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