Tagged Questions
4
votes
1answer
46 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
94 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
32 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
68 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 ...
2
votes
0answers
37 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 ...
5
votes
2answers
106 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
181 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
80 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
84 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
159 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
146 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
249 views
Ambient Isotopy
From Hatcher's (edit. 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 ...
2
votes
1answer
105 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 ...
