In this paper we show the existence of a pair of commuting completely nonunitary contractions
(
S
,
T
)
(S,T)
on a Hilbert space, whose joint Taylor spectrum contains the torus, such that there is a bounded analytic function
h
h
on the bidisk with
h
(
S
,
T
)
=
0
h(S,T) = 0
.