Given a number
β
>
1
\beta > 1
, the beta-transformation
T
=
T
β
T =T_{\beta }
is defined for
x
∈
[
0
,
1
]
x \in [0,1]
by
T
x
:=
β
x
Tx := \beta x
(mod 1). The number
β
\beta
is said to be a beta-number if the orbit
{
T
n
(
1
)
}
\{T^{n}(1)\}
is finite, hence eventually periodic. In this case
β
\beta
is the root of a monic polynomial
R
(
x
)
R(x)
with integer coefficients called the characteristic polynomial of
β
\beta
. If
P
(
x
)
P(x)
is the minimal polynomial of
β
\beta
, then
R
(
x
)
=
P
(
x
)
Q
(
x
)
R(x) = P(x)Q(x)
for some polynomial
Q
(
x
)
Q(x)
. It is the factor
Q
(
x
)
Q(x)
which concerns us here in case
β
\beta
is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether
Q
(
x
)
Q(x)
must be cyclotomic in this case, particularly if
1
>
β
>
2
1 > \beta > 2
. We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in
[
1
,
1.9324
]
∪
[
1.9333
,
1.96
]
[1,1.9324]\cup [1.9333,1.96]
(an infinite set), by a search up to degree
50
50
in
[
1.9
,
2
]
[1.9,2]
, to degree
60
60
in
[
1.96
,
2
]
[1.96,2]
, and to degree
20
20
in
[
2
,
2.2
]
[2,2.2]
. We find the smallest counterexample, the counterexample of smallest degree, examples where
Q
(
x
)
Q(x)
is nonreciprocal, and examples where
Q
(
x
)
Q(x)
is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to
2
2
from above, and infinite sequences of
β
\beta
with
Q
(
x
)
Q(x)
nonreciprocal which converge to
2
2
from below and to the
6
6
th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in
[
1
,
2
]
[1,2]
. The Pisot numbers for which
Q
(
x
)
Q(x)
is cyclotomic are related to an interesting closed set of numbers
F
\mathcal {F}
introduced by Flatto, Lagarias and Poonen in connection with the zeta function of
T
T
. Our examples show that the set
S
S
of Pisot numbers is not a subset of
F
\mathcal {F}
.