1
vote
1answer
30 views

Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
1
vote
1answer
137 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
2
votes
2answers
72 views

the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
0
votes
0answers
19 views

Subsets of immersions that are embeddings

Let $X$ be a manifold and let $Y$ be a submanifold, possibly with boundary. I am dealing with a situation where $f:X\to \mathbb{R^3}$ is an immersion, but $f\vert_Y:Y\to \mathbb{R}^3$ is an embedding. ...
1
vote
1answer
55 views

Moves for regular homotopies of immersions of $S^1$ in the plane

What is a set of moves to combinatorially describe regular homotopies of (smooth) immersions $S^1\to \mathbb R^2$?
3
votes
1answer
41 views

Unknotting number formally?

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings ...
0
votes
0answers
55 views

What is the type of $u$ in this definition of knot?

I am going to quote from the second paragraph of the Introduction of Möbius Energy of Knots and Unknots by Michael H. Freedman, Zheng-Xu He and Zhenghan Wang. Let $\gamma = \gamma (u)$ be a ...
2
votes
1answer
36 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
1
vote
1answer
160 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
2
votes
1answer
74 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
4
votes
1answer
81 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
4
votes
1answer
64 views

How to make a $C^1$ knot into a $C^\infty$ knot

Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
1
vote
0answers
114 views

Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
1
vote
1answer
114 views

A question about orthogonal projections of Smooth Embeddings of the circle.

Question: Let $f$ be a smooth embedding of $S^1\rightarrow \mathbb{R}^3$. Given an element $v\in S^2$ we have the orthogonal projection $\pi_v:\mathbb{R}^3\rightarrow P_v$ to the plane $P_v$ = the ...
1
vote
1answer
129 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
3
votes
0answers
43 views

Linking integral unchanged over continuous deformations

Say we first have two curves, $C_1$ and $C_2$ which are knotted together. Let $C_2'$ be a continuous deformation of $C_2$ such that $C_2$ does not cross $C_1$ as it is deformed into $C_2'$. How ...
6
votes
2answers
118 views

How does smoothness prevent “singularities”?

This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. ...
5
votes
0answers
269 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
2
votes
1answer
112 views

How to detect a twist or framing in a 3-manifold.

This question is somewhat a continuation of the question Gluing a solid torus to a solid torus with annulus inside. If we consider a genus one handlebody $U$ with an (nontwisted)annulus inside the ...
1
vote
0answers
87 views

Embedding of $T^{2}$ in $S^{1}\times S^{2}$.

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map and $i_{*}:\pi(T^{2})\rightarrow\pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times ...
3
votes
1answer
270 views

Surgery, framing and Dehn twist

Let $L$ be a framed knot in $S^3$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$. How can I interpretate the following sentence? "We identify $U$ with $S^1 \times B^2$ so that $L$ is ...
3
votes
1answer
245 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
3
votes
0answers
368 views

Ambient Isotopy

From Hirsch's Differential Topology, p. 180. The first of the isotopy extension theorems says; Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an isotopy of $A$. If ...
2
votes
1answer
160 views

Locally flat submanifold

Recently I found the following definition: Let $M^{n}$ be an $n-$dimensional topological manifold. Then $N^{k}\subseteq M^{n}$ is a locally flat submanifold if for every $x\in N$ there exists an open ...