We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with
n
n
twists, namely
Γ
n
=
⟨
x
,
y
|
[
x
n
,
y
]
=
1
⟩
{\Gamma }_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle
into the group
S
U
(
r
)
SU(r)
. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case
r
=
2
r = 2
, we provide a complete geometric description of the character variety, proving that this
S
U
(
2
)
SU(2)
-character variety is a deformation retract of the larger
S
L
(
2
,
C
)
SL(2,\mathbb {C})
-character variety, as conjectured by Florentino and Lawton. In the case
r
=
3
r = 3
, we also describe different strata of the
S
U
(
3
)
SU(3)
-character variety according to the semi-simple type of the representation.