All Questions
Tagged with 2-knots knot-theory
18 questions
-3
votes
1
answer
106
views
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
6
votes
1
answer
148
views
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
7
votes
0
answers
170
views
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 ...
6
votes
0
answers
118
views
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
11
votes
2
answers
450
views
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
3
votes
1
answer
778
views
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
views
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
5
votes
0
answers
251
views
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
8
votes
1
answer
570
views
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
votes
0
answers
320
views
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
921
views
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
24
votes
5
answers
3k
views
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
answers
2k
views
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
10
votes
2
answers
910
views
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
4
votes
4
answers
1k
views
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
0
votes
1
answer
415
views
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
11
votes
2
answers
2k
views
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
8
votes
5
answers
1k
views
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