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Questions tagged [kirby-diagram]

25 questions
6
votes
3 answers

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 type of the following diagram?
user302934
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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 coefficient -1. (which is a standard exercise in geometric…
cjackal
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5
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2 answers

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 and 3 handle cancelling pair to reduce the Kirby…
Rob Smith
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1 answer

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 structure, etc.). This raises a couple of…
Willem Noorduin
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4
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1 answer

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 the 3- (and 4-) handles that close the…
Turion
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4
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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 one 1 handle? Can we still compute its intersection…
horned-sphere
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4
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1 answer

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 other negative. In other words, the writhe of the…
Overflowian
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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 construct a surgery diagram where the link has all…
Overflowian
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4
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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 and page 116) Now, we consider compact…
T.O.
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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 diagram, adds one 1-handle and connects it to the…
Turion
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3
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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 when $n > 4$. Further, I do not know whether you can…
Willem Noorduin
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3
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Calculating homology of cobordism of 3-manifolds from Kirby diagram

I've been reading Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci and Stipsicz, and have gotten stuck on the following exercise on p. 44. Below $Y_1, Y_2$ are closed 3-manifolds, and $Q$ denotes intersection form: It is only a little…
Hrhm
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3
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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 thought at first he is referring to the fact that…
horned-sphere
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3
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3 answers

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 \dots L_n $ be one such link in $ S^3$ for $M$,…
Subhankar D.
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2 answers

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 foundation in undergraduate analysis, working through…
kingdras
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