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In this paper we discuss the dynamics as well as the structure of theparameter space of the one-parameter family of rational maps $\dsf_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}$ with free critical orbit$\pm\sqrt{2}\xrightarrow{(2)}0\xrightarrow{(4)}t\xrightarrow{(1)}...$. Inparticular it is shown that for any escape parameter $t$ the boundary of thebasin at infinity $\A_t$ is either a Cantor set, a curve with infinitely manycomplementary components, or else a Jordan curve. In the latter case the Juliaset is a Sierpi\'nski curve.

On The Dynamics Of The Rational Family $\mathbf{f_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}}$