  We determine the Thurston unit ball of a family of $n$-chained link, denoted
by $C(n,p)$, where $n$ is the number of link components and $p$ is the number
of twists. When $p$ is strictly positive, we prove that the Thurston unit ball
for $C(n,p)$ is an $n$-dimensional cocube, for arbitrary $n$. Moreover, we
clarify the condition for which $C(n,p)$ is fibered and find at least one
fibered face for any $p$. Finally we provide the Teichm\"uller polynomial for
the face of Thurston unit ball of $C(n, -2)$ with $n\geq 3$.
