Example of Codimension-1 Manifolds that are not Isotopic.?

[Bacle]

Hi, Everyone:

A question on knots, please; comments,references

appreciated. The main points of confusion are noted

with a ***:


1)I am trying to understand how to describe the knot

group Pi_1(S^3-K) as a handlebody ( this is not the

Wirtinger presentation; this is from some old notes

(which are not too clear now). This is what I have

so far:


We clearly start by embedding a thickened

K in S^3 as e(K). ( I imagine there is an assumed

orientation, but I can't tell where it comes from).

Topologically, S^3 is clearly the compactification

of R^3.

We then have a graph-theoretic description of e(K)

in terms of edges {e_i} and vertices {v_i}.


We then define a representation of the knot

as a handlebody (***) by setting up the equivalences:


0-handles <--> Closed neighborhoods of 00

1-handles <--> Holes in the embedding e(K)

2-handles <--> crossings in e(K)


We take the above equivalence to define relations

on the faces :(***)




The generators of Pi_1(S^3-K) are given by the

edges e_i of the knot, and the relations are given

by applyng the del. operator on each of the faces.


Thanks For any Suggestions.

 

[Bacle]

My Bad: I mixed up two posts. Please Ignore the Title.

 

[Bacle]

Never Mind, I got it.