All Questions
Tagged with surgery-theory knot-theory
26 questions
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Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 593
6
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1
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472
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4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere
I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation:
Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
- 335
3
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86
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Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s
Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...
- 335
8
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Surgery along knots and connected sum
Denote $S^3_{p/q}(K)$ by performing $p/q$-surgery along a knot $K$ in $S^3$.
Let $K$ and $J$ be two arbitrary oriented non-trivial knots in $S^3$. Is there a nice relation between surgery on the ...
- 213
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377
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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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Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
The Two Summands Conjecture states that surgery ...
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Bipartedly slice links and their surgeries
A link L in $S^3$ is said to be strongly slice if $L=∂D$,where $D$ is a disjoint union of smoothly and properly embedded disks in $B^4$.
A link $L$ in $S^3$ is called bipartedly slice if $L = L_1 \cup ...
user160180
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248
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Concordance, surgery and homology cobordism
In this post, we discuss the relation between the concordance of knots in $S^3$ and the integral homology cobordism.
Following its notation, assume that knots $K_0$ and $K_1$ in $S^3$ are concordant. ...
user160180
5
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1
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433
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Dehn surgery on $S^3$ along a Hopf link with rational surgery coefficients
Is there an exhaustive list of conditions satisfied by rational surgery coefficients assigned to the components of the Hopf link in $S^3$ such that the resulting 3-manifold by Dehn surgery acting on $...
8
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Integer surgeries along links yielding lens spaces
Does there exist an integer $N$ such that any lens space $L(p,q)$ can be obtained by integer surgery from $S^3$ along a link $L$ with at most $N$ components?
EDIT:
I have worked out the comment by ...
- 1,314
5
votes
2
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329
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Negative surgeries on negative knots
This question is two-fold.
The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of ...
- 1,430
1
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1
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469
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Integer surgery on $S^3$
I know that any compact orientable 3-manifold can be obtained from the three sphere $S^3$ by an integer surgery. I am not sure why the surgery operation is completely determined by Where we map ...
- 504
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2
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522
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Rational surgery and attaching $2$-handles
It is well-known fact that integral Dehn surgeries on $3$-sphere $S^3$ are viewed as the result on the boundary of attaching $2$-handles $B^2 \times B^2$ to the $4$-ball $B^4$.
Is there an analogue ...
user150450
7
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1
answer
862
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What are these 3-manifolds from surgery?
I know that surgery on the unlink with +0 slope gives $S^2 \times S^1$ (where all the links above are embedded in $S^3$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $S^...
- 187
4
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2
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360
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Chirality and Anti-Chirality of links in 3 and in 5 dimensions
We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory:
https://en.wikipedia.org/wiki/Chiral_knot
My ...
- 10.5k
2
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1
answer
166
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What is the most symmetric configuration of four 2-surfaces linked in $S^4$?
What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$?
To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
- 10.5k
3
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0
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106
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A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
- 10.5k
3
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192
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Surgery to unlink $S^p$ and $S^q$ in $S^d$
We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres.
I am looking for a surgery to unlink $S^p$ and $...
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The relations between some 3-components links and trefoil knots [closed]
It is intuitive to see that the 3-components links (under Alexander–Briggs notations) $6^3_1, 6^3_2, 6^3_3$ are closely related to each other; in a sense by doing a cut-gluing or sew-gluing surgery, ...
- 10.5k
2
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1
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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 ...
user22741
6
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1
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547
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Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1
I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial_{-}M, \partial_{+}M)$ in the previous question breifly. Framings in the ...
user22741
3
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1
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581
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Framings in the definition of Reshetikhin-Turaev TQFT
I posted the following question at Mathe Stack Exchange.link text But it has not yet answered. I am sorry if you check both sites but I also want people here to look at this problem.
I am studying ...
user22741
0
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199
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Finding a ribbon graph for a mapping class group action
Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is ...
- 1
2
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1
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332
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A special ribbon graph presents a cylinder.
I am reading "Quantum Invariants of Knots and 3-Manifolds" by Turaev. I have a dificulty to understand the proof of Lemma 2.6 on page 172.
The lemma says that a special ribbon graph drawn on page 167 ...
- 93
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2
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703
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Do the results of (1/n)-surgery determine the link?...
Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. ...
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55
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3
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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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