Graphics Reference
In-Depth Information
Sometimes it is convenient to consider knots in
S
3
rather than
R
3
because
S
3
is a
compact space, but there is again no real difference in the theory since, using the
stereographic projections,
S
3
can be thought of as just
R
3
with one point added. Note
also that, since all knot, are homeomorphic to
S
1
, classifying them is not a question
of determining if they themselves are homeomorphic because they are. What makes
knots different is their imbedding in
R
3
. Every knot in the plane is necessarily trivial
by the Schoenflies theorem.
In order not to have to deal with wild imbeddings, one also usually assumes that
knots are polygonal.
Definition.
Let
K
be a knot. The fundamental group p
1
(
R
3
-
K
) is called the
group
of the knot
K
. (The base point of the fundamental group was omitted because we are
only interested in the group up to isomorphism.)
The group of a knot plays a large role in the study of knots but does not deter-
mine the knot completely because there exist inequivalent knots that have the same
knot group, such as for example, the
square knot
and the
granny knot
shown in Figure
7.22. Certainly, equivalent knots have isomorphic knot groups because their comple-
ments are homeomorphic. The knot group is only one of many interesting invariants
associated to a knot.
Before we list a few important known facts about the classification of knots, we
define a well-known infinite family of knots that serve as useful examples.
Definition.
A
torus knot of type
(
p,q
), where p and q are relatively prime, is a knot
that can be imbedded in a torus and has the property that it cuts a meridian circle of
the torus in p points and a circle of latitude in q points. In cylindrical coordinates, a
specific instance of such a knot is the curve
(
)
r
=+
2
cos
q
q
p
(
)
z
=
sin
q
q
p
that lies in the torus in
R
3
(the circle in the x-z plane with center (2,0,0) and radius
1 rotated about the z-axis) defined by the equation
2
(
)
2
r
-
2
+=
z
1
.
square knot
granny knot
Figure 7.22.
Two inequivalent knots with isomorphic knot groups.
