All Questions
24 questions
37
votes
0
answers
2k
views
What is the three-dimensional hyperbolic volume of a four-manifold?
Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
...
- 10.5k
25
votes
0
answers
1k
views
Concordance and homology cobordism
If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if $...
- 779
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
17
votes
1
answer
1k
views
Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?
Let $K$ be a knot in the 3-sphere $S^3$.
Here we denote by $s(K)$ Rasmussen's s-invariant for $K$,
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a $2$-...
- 391
12
votes
2
answers
308
views
Property P and R for general 3-manifolds
Let $Y$ be a closed oriented $3$-manifold and $K$ be a knot in $Y$. We say $K$ is the unknot if $K$ is contained in a local $3$-ball in $Y$ and is unknotted therein.
Generalized Property R:
If a Dehn ...
- 557
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
9
votes
2
answers
546
views
Rational slice knot that is not slice
Does there exists a knot $K\subset \mathbb{S}^3$ such that
$K$ is not slice
$\exists W^4$, $\partial W = \mathbb{S}^3$ rational homology ball
$\exists $ properly embedded smooth disk $(D,\partial D)\...
- 2,533
9
votes
2
answers
457
views
A knot in the solid torus and a Mazur manifold
Part 1: The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130:
He indicates that the knot drawn in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \...
- 213
9
votes
2
answers
351
views
Gordon's approach: slice knots and contractible $4$-manifolds
Let $K \subset S^3$ be a slice knot. Then it bounds a smooth embedded disk $D \subset B^4$. Let $S^3_{p/q}(K)$ denote a $3$-manifold obtained by $p/q$-surgery on $K \subset S^3$.
The following theorem ...
- 606
9
votes
1
answer
627
views
Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively
I'm interested in a complexity question related to problems like the slice-ribbon problem.
To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...
- 44.4k
8
votes
0
answers
445
views
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
7
votes
2
answers
361
views
Double ($p$-fold) coverings of $B^4$ along ribbon/slice disks
I have two questions that seem to be related.
I wonder if there is a user-friendly algorithm (starting from ribbon/slice presentation of knots/disks) for the construction of double (in general $p$-...
- 606
6
votes
2
answers
395
views
Slice knots in 3-manifolds
Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$...
- 557
6
votes
1
answer
324
views
Computation of $\pi_1$ for a Mazur manifold and its boundary
If we attach a $4$-dimensional $1$-handle $D^1 \times D^3$ to a $4$-dimensional $0$-handle $B^4$, we obtain $S^1 \times D^3$. The null homologous knot in $S^1 \times S^2$ indicated in the picture ...
- 606
6
votes
1
answer
149
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
5
votes
1
answer
340
views
0-surgery on a fibered hyperbolic ribbon knot
Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered?
I tried looking at ...
- 1,174
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
4
votes
2
answers
544
views
$0$-surgery of slice knots and contractible manifolds
We know that if we attach $4$-dimensional $2$-handle $D^2 \times D^2$ to $S^1 \times S^2$, then we produce a contractible $4$-manifold. In this case, $S^1 \times S^2$ is $0$-surgery on the unknot.
If ...
user160180
4
votes
1
answer
359
views
Mazur homology spheres
This is Example 6.47 in Saveliev's book Invariants for homology $3$-spheres:
Let us consider a two-component link $\mathcal L = L_1 \cup L_2$ in
$S^3$ such that $\mathrm{lk}(L_1,L_2) = \pm 1$ and the ...
user160180
3
votes
1
answer
212
views
Picturing twisting of strands explicitly after blow downs
In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
- 213
3
votes
0
answers
91
views
Handlesliding a two component, linking number 1 link
Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...
- 5,349
2
votes
0
answers
201
views
The Kirby diagram of a manifold glued along the lens space $L(p,1)$
Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
- 673
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
0
votes
0
answers
114
views
May this slice disk for the unknot be pushed into the boundary?
Write the 4-ball as $\mathbb{D}^4=\mathbb{D}^2\times \mathbb{D}^2$.
Then its boundary $\mathbb{S}^3\simeq \mathbb{S}^1\times \mathbb{D}^2\cup \mathbb{D}^2\times \mathbb{S}^1$. We will use implicitely ...
- 2,533