Translating annuli for mean curvature flow

We construct a family $A$ of complete, properly embedded, annular translators M such that M lies in a slab and is invariant under reflections in the vertical coordinate planes. For each M in $A$, M is asymptotic as $z\to -\infty$ to four vertical planes $\{y=\pm b\}$ and $\{y=\pm B\}$ where $0<b\le B<\infty$. We call b and B the inner width and the (outer) width of M. We show that for each $b\ge \pi /2$ and each $s>0$, there is an $M\in A$ with inner width b and with necksize s. (We also show that there are no translators with inner width $<\pi /2$ having the properties of the examples we construct.)