Graphics Reference
In-Depth Information
Figure 7.23.
A torus knot of type (3,5).
Figure 7.23 shows an example of a torus knot.
7.4.1.17. Theorem.
(1) A tame knot is trivial if and only if the group of the knot is infinite cyclic (iso-
morphic to
Z
). There is an algorithm that determines whether or not a knot
is trivial.
(2) Two tame knots have homotopy equivalent complements if and only if their
knot groups are isomorphic. (Conjecture: If two tame knots have homeomor-
phic complements, then they have the same knot type.)
(3) There exist infinitely many knot types. For example, the torus knots of type
(p,q) are all inequivalent.
(4) The abelianization of every knot group is infinite cyclic.
(5) If
K
is a tame knot, then p
i
(
R
3
-
K
) = 0 for i > 1.
Proof.
The proofs of most of these facts are much too complicated to give here. See
the references for knot theory listed earlier.
7.4.2
Covering Spaces
The topic of this section is intimately connected with the fundamental group but also
has important applications in other areas such as complex analysis and Riemann sur-
faces. Section 8.10 in the next chapter will continue the discussion and discuss the
related topic of vector bundles.
We begin with some basic terminology and motivational remarks. See Figure 7.24.
Definition.
A
bundle over a space
X
is a pair (
Y
,p), where
Y
is a topological space
and p :
Y
Æ
X
is a continuous surjective map. One calls
Y
the
total space
, p the
pro-
jection
, and
X
the
base space
of the bundle. The inverse images p
-1
(
x
) Õ
Y
for
x
Œ
X
,
are called the
fibers
of the bundle.
In our current context we should think of the total space of a bundle as consist-
ing of a union of fibers that are glued together appropriately. Of course, the general
case of an arbitrary surjective map p does not lead to anything interesting. The inter-
esting case is where all the fibers are homeomorphic to a fixed space
F
. The obvious
example of that is the product of the base space and
F
.
Definition.
A bundle over
X
of the form (
X
¥
F
,p), where p is the projection onto
the first factor defined by p(
x
,
f
) =
x
, is called the
product bundle with fiber F
.
