Questions tagged [2-knots]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
2answers
167 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 ...
5
votes
0answers
99 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 ...
10
votes
2answers
311 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 ...
3
votes
1answer
268 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 $J(x,t)=(\phi_t(x),t)$. ...
6
votes
1answer
378 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?
2
votes
0answers
251 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 ...
5
votes
0answers
177 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 $...
6
votes
1answer
381 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 ...
7
votes
0answers
277 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 ...
13
votes
2answers
867 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 ...
6
votes
1answer
815 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}_{\...
22
votes
5answers
2k 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 ...
10
votes
3answers
892 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 ...
18
votes
3answers
1k 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 ...
9
votes
2answers
736 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, -\...
4
votes
4answers
996 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 ...
0
votes
1answer
339 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 ...
11
votes
2answers
1k 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 ...
7
votes
5answers
948 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-...
8
votes
2answers
797 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. ...