Bacle
- 656
- 1
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.
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.