5
votes
1answer
123 views

Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question: If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...
3
votes
2answers
137 views

Handlebody decomposition of a 3-manifold adapted to a link

Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that: ...
4
votes
2answers
145 views

Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed ...
1
vote
2answers
273 views

Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
16
votes
3answers
374 views

Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...
7
votes
2answers
165 views

How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group which can be generated by n meridians, but not less than n, is an n-bridge knot. There is a one-sentence update, saying that ...
1
vote
1answer
128 views

Two links with the same signatures but unknown if they are related by Kirby moves

I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that: the signatures of $L_1, L_2$ are known. we do not know if they are related by Kirby moves. If so, could you specify the ...
9
votes
1answer
334 views

What are the homotopy classes of two-component links in $\mathbb{RP}^3$?

This question comes from an unanswered question on Math Stack Exchange. A two-component link in $\mathbb{RP}^3$ is any embedding $S^1\uplus S^1\to \mathbb{RP}^3$. Two such links are homotopic if ...
1
vote
0answers
83 views

Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
6
votes
1answer
259 views

integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...
6
votes
2answers
251 views

Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows. Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components? Here is what I mean: The Borromean rings form a famous link $B$ (a smooth ...
4
votes
1answer
195 views

Equivalent to Oriented knot complement conjecture

I would like to see why the following two statements in Kirby's list of problem are equivalent: Statement 1: If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are ...
9
votes
1answer
352 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 ...
25
votes
0answers
897 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. ...
5
votes
1answer
519 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
3
votes
1answer
301 views

A question about Dehn filling in the unknot.

Hello, If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ ...
7
votes
2answers
417 views

Do the results of (1/n)-surgery determine the link?…

Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. ...
9
votes
1answer
305 views

Ramified cover of 3-ball

I it true that any connected oriented 3-manifold with nonempty boundary can be obtained as a ramified cover of 3-ball with a ramification at a link? link = a 1-dimensional submanifold with possibly ...
8
votes
1answer
501 views

Lower bound on number of tetrahedra needed to triangulate a knot complement

Following along a similar line to the question asked here: Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume V? Let $K$ be a (hyperbolic) ...
5
votes
2answers
347 views

Unknotting tunnels in surface bundles

Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$. In Sakuma's survey article The ...
5
votes
2answers
540 views

Intutive interpretation about Linking forms

Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with $H_{*}(M;Q)=H_{*}(S^3;Q)$ As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's ...
15
votes
2answers
1k views

Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ...
20
votes
3answers
1k views

How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]

With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about ...
36
votes
3answers
2k views

Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
8
votes
0answers
222 views

Krull rings and determinantal invariants

During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
3
votes
2answers
408 views

SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...