Questions tagged [2-knots]

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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 ...
Alessio Di Prisa's user avatar
7 votes
0 answers
151 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 ...
Daniel H. Hartman's user avatar
3 votes
2 answers
237 views

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 ...
Matt Cuffaro's user avatar
6 votes
0 answers
115 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 ...
Román Aranda's user avatar
11 votes
2 answers
436 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 ...
Henry's user avatar
  • 1,410
3 votes
1 answer
675 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 $J(x,t)=(\phi_t(x),t)$. ...
Joao Faria Martins's user avatar
6 votes
1 answer
431 views

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?
silvascientist's user avatar
3 votes
0 answers
339 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 ...
Zhiyun Cheng's user avatar
5 votes
0 answers
241 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 $...
Blake's user avatar
  • 1,025
8 votes
1 answer
520 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 ...
Mark Hughes's user avatar
7 votes
0 answers
309 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 ...
Scott Carter's user avatar
  • 5,234
13 votes
2 answers
1k views

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 ...
Ryan Budney's user avatar
  • 42.8k
6 votes
1 answer
897 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}_{\...
John Pardon's user avatar
  • 18.3k
23 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 ...
Theo Johnson-Freyd's user avatar
11 votes
3 answers
1k views

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 ...
John Vrem's user avatar
  • 374
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 ...
Andrew Lobb's user avatar
10 votes
2 answers
879 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, -\...
Andrew Lobb's user avatar
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 ...
Megan's user avatar
  • 990
0 votes
1 answer
403 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 ...
Megan's user avatar
  • 990
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 ...
Ryan Budney's user avatar
  • 42.8k
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-...
Scott Carter's user avatar
  • 5,234
10 votes
2 answers
887 views

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. ...
Scott Morrison's user avatar