Tagged Questions
0
votes
0answers
41 views
Density of continuous knots in the plane transversal to some circles
This is an exercise from the book "Knots and Links" by Rolfsen (exercise 6 in section 2C)
Let $\kappa : S^1 \rightarrow \mathbb{R}^2-(0,0)$ be a continuous imbedding. Let $M := \{ x \in ...
2
votes
1answer
52 views
Annulus Theorem
I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is
Use the Schoenflies theorem ...
1
vote
0answers
24 views
seek visual pictures or video on decomposition of manifolds
In my study of knot theory, I notice that I lack examples to show some classical decomposition theorems in 3-dimensional manifolds, such as JSJ decomposition theorem, Milnor's prime decomposition ...
1
vote
1answer
70 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
2answers
136 views
What is smoothness needed for?
We can either define a knot to be
(1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$
or
(2) a piecewise linear, simple closed curve in $\mathbb R^3$
Then these two definitions are ...
8
votes
0answers
174 views
Why is the knot group of the trefoil isomorphic to the group of 3-braids?
I apologise in advance for the vagueness of this question but I have not been able to find very much info on the topic and have made very little progress on my own.
I am trying to understand why the ...
2
votes
1answer
65 views
Remove links by Kirby moves
I am trying to prove the following proposition.
proposition; If in a framed link $L$ a component $K$ is an unknot with framing zero which links only one other component $H$ geometrically once, ...
1
vote
1answer
102 views
Are satellite knots prime?
Which satellite knots are prime?
I do know that connected sum of knots is a satellite operation, but I found this statement:
"the satellite knots all have structures which are well known and ...
