Let
n
>
m
n>m
be fixed positive coprime integers. For
v
>
0
v>0
, we give a topological description of the set
Λ
(
v
)
\Lambda (v)
, consisting of points
[
x
:
y
:
z
]
[x:y:z]
in the complex projective plane for which the equation
x
ζ
n
+
y
ζ
m
+
z
=
0
x\zeta ^n +y \zeta ^m+z=0
has a root with norm
v
v
. It is shown that the set
Ω
(
v
)
=
P
C
2
∖
Λ
(
v
)
\Omega (v)= {\mathbb P_{\mathbb C}} ^2 \setminus \Lambda (v)
has
n
+
1
n+1
components. Moreover, the topological type of each component is given. The same results hold for
Λ
\Lambda
and
Ω
=
P
C
2
∖
Λ
\Omega ={\mathbb P_{\mathbb C}}^2 \setminus \Lambda
, where
Λ
\Lambda
denotes the set obtained as the union of all the complex tangent lines to the
3
3
-sphere at the points of the torus knot, that is, the knot obtained by intersecting
{
[
x
:
y
:
1
]
∈
P
C
2
:
|
x
|
2
+
|
y
|
2
=
1
}
\{[x:y:1] \in \mathbb {P}_{\mathbb C}^2 : |x|^2+|y|^2=1\}
and the complex curve
{
[
x
:
y
:
1
]
∈
P
C
2
:
y
m
=
x
n
}
\{[x:y:1] \in {\mathbb P_{\mathbb C}} ^2 : y^m=x^n\}
. Finally, we use the linking number of a distinguished family of circles and the torus knot to give a numerical invariant which determines the components of
Ω
\Omega
in a unique way.