Questions tagged [2-knots]
The 2-knots tag has no usage guidance.
24 questions
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Can surfaces be interestingly knotted in five-dimensional space?
It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves ...
- 54.6k
19
votes
3
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topological "milnor's conjecture" on torus knots.
Here's a question that has come up in a couple of talks that I have given recently.
The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly ...
- 653
13
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2
answers
1k
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Explicit embeddings of Cappell-Shaneson knots
In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare ...
- 44.3k
11
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2
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Is there a known invariant for knotted surfaces defined by skein relations?
Is there a known invariant for knotted surfaces in $\mathbb{R}^4$ (possibly with additional structures, e.g. colored, framed, etc.) which can be defined using skein relations? By skein relations for ...
- 1,430
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3
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Alexander polynomial or Reidemeister torsion for knotted surfaces?
An important invariant of a knot in $S^3$ is its Alexander polynomial, related also to Reidemeister torsion. Is there something like that for knotted surfaces in $S^4$? If not, what are the ...
- 374
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2
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slice-ribbon for links (surely it's wrong)
The slice-ribbon conjecture asserts that all slice knots are ribbon.
This assumes the context:
1) A `knot' is a smooth embedding $S^1 \to S^3$. We're thinking of the 3-sphere as the boundary of ...
- 44.3k
10
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2
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Has anyone tabulated 2-knots? Would anyone like to try?
I'd love to have a list of 'small' $2$-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates
Write a movie presentation, and count the frames.
...
- 7,800
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votes
2
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slice=ribbon generalization to higher genus + potential counterexamples to slice=ribbon.
I have two questions about the slice=ribbon conjecture.
(1) If a knot $K \hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 \times [0, -\...
- 653
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5
answers
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Braided Monoidal 2-categories with duals
Which categorifications give explicit braided monoidal 2-categories with duals?
This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2-...
- 5,264
8
votes
1
answer
570
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Differences between various categories of surface embeddings in 4-space
This is a very naive question, but I'm trying to understand the difference between the various categories when it comes to embedding surfaces in 4-dimensional manifolds. The situation I'd really like ...
- 123
7
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0
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170
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Does the non-cancelation theorem hold for 2-knots?
In Rolfsen's knots and links, he shows that, as a consequence of the unknotting theorem, that if you connect sum two knots and get the unknot, they both had to be unknotted. Does the same statement ...
7
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0
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320
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What is the historical connection between Zeeman's twist spinning and Fox's Examples?
Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...
- 5,264
6
votes
1
answer
441
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Parameterization of a knotted surface?
I'm looking for a parameterization $(x_1(u,v),x_2(u,v),x_3(u,v),x_4(u,v))$ of a knotted sphere in $\mathbb R^4$. How might one go about finding such a parameterization?
- 162
6
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1
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Is the complete functorial structure for Khovanov--Lee homology known?
I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$.
The groups $\operatorname{Kh}_{\...
- 18.7k
6
votes
1
answer
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Knotted concordances of slice links
Are there any examples of a link $L$ such that:
$L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap ...
- 193
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0
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Is there a notion of tunnel number for 2-knots?
Given an embedded circle $K$ in $S^3$, the tunnel number of $K$ is the minimum number of embedded arcs one needs to add to $K$ so that the complement of $K$ and the arcs is a handlebody.
For an ...
- 121
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Ribbon knot presentations
Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and $...
- 1,025
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votes
4
answers
1k
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4-genus of a 2-bridge link
How can we calculate the 4-genus of a link L? The 4-genus is defined to be the minimal genus of orientable surface bounded by L in B^4. Is there any routine method to calculate that?
Especially, any ...
- 1,040
4
votes
1
answer
226
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Equivalence of knotted spheres in $S^4$
Say we have two smoothly embedded spheres $K, K' \subset S^4$ that are equivalent in the sense that there is a diffeomorphism of pairs $(S^4, K)$ and $(S^4, K')$. Does it follow that they are ...
- 41
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2
answers
244
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Is the Artin Spin construction related to the suspension functor?
I've been reading about the Artin Spin operation. It's defined as taking the classical $n$-knot ($S^n\hookrightarrow S^{n+2}$) to an $(n+1)$-knot. For the $1$-knot case (in $\mathbb{R}^3$), I ...
- 429
3
votes
1
answer
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Isotopy extension theorem: how non-unique is ambient isotopy
Let $M$ and $N$ be smooth manifolds. Consider an isotopy of $M$ inside $N$. This means that we have a level preserving embedding $J\colon M\times [0,1] \to N \times [0,1]$. Put $J(x,t)=(\phi_t(x),t)$. ...
3
votes
0
answers
374
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Jones polynomial of 2-knots
Question: is it possible to define the Jones polynomial for knotted surfaces (or $S^2$ for simplicity) in $R^4$?
Jones polynomial has several definitions (see How many definitions are there of the ...
- 517
0
votes
1
answer
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If the 4-genus of a link is zero, is it a slice link?
An n-component slice link is a link that bounds n disjoint discs in B^4. And the 4-genus of a link is defined to be the minimal genus of orientable surfaces bounded by it in B^4.
My question is: if ...
- 1,040
-3
votes
1
answer
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Knot group of mirror image [closed]
Are the knot group and the knot group of its mirror image isomorphic?
And,How about the case of knotted surfaces?
- 1