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wofsy
Nov3-09, 08:17 PM
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
Nov5-09, 12:35 AM
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
Nov5-09, 11:11 AM
Look at an unknot and a trefoil knot and compare the fundamental group of the complement.
wofsy
Nov5-09, 12:38 PM
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
Nov5-09, 11:34 PM
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
Nov6-09, 07:01 AM
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?