To investigate the degree
d
d
connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied
σ
d
\sigma _d
-invariant laminations, where
σ
d
\sigma _d
is the
d
d
-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials
f
(
z
)
=
z
2
+
c
f(z) = z^2 +c
. In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials
f
λ
(
z
)
=
z
3
+
λ
2
z
f_\lambda (z)=z^3+\lambda ^2 z
in a series of three articles. In the present paper, the first in the series, we construct a lamination
C
s
C
L
C_sCL
together with the induced factor space
S
/
C
s
C
L
\mathbb {S}/C_sCL
of the unit circle
S
\mathbb {S}
. As will be verified in the third paper of the series,
S
/
C
s
C
L
\mathbb {S}/C_sCL
is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.