Question whether 2 spaces are diffeomorphic

[wofsy]

Suppose I have two embeddings of the circle into the 3 sphere. Is S3-minus the first image diffeomorphic to S3 - the image of the second?

 

[Office_Shredder]

The answer is no. This is actually a knot theory concept... an embedding of S¹ in R³ is a knot, which are all homeomorphic to each other (generally taken that a knot is a homeomorphic image of S¹ in R³. But in R³, there may not be a homeomorphism of R³ to itself that maps one to the other. There are some classic examples of knots that are not equivalent (knots tend to be embeddings) but I don't remember any off the top of my head, I'm sure a google search will reveal them

 

[zhentil]

Look at an unknot and a trefoil knot and compare the fundamental group of the complement.

 

[wofsy]

Look at an unknot and a trefoil knot and compare the fundamental group of the complement.

Thanks I will do tyhat.

It is interesting because the first homology is just Z - by Alexander duality or a Meyer-Vietoris sequence argument.

 

[zhentil]

If I'm getting this right, the fundamental group of the complement of a non-trivial knot should be a bouquet of circles-type situation; i.e. you have some generators with some anti-commutation relations, and the commutator kills all of it.

 

[wofsy]

If I'm getting this right, the fundamental group of the complement of a non-trivial knot should be a bouquet of circles-type situation; i.e. you have some generators with some anti-commutation relations, and the commutator kills all of it.

That has to right. And there is no torsion in the group mod its commutator subgroup.
Do you know the generqator and relations for one of these?