  We model the sub-millimetre polarization patterns that are expected for
filamentary clouds that are threaded by helical magnetic fields. We previously
developed a three parameter model of such clouds (Fiege & Pudritz 2000a), which
are described by a concentration parameter $C$, and two flux to mass ratios
$\Gz$ and $\Gphi$ to specify the mass loading of the poloidal and toroidal
field lines respectively. Our models provide a simple and purely geometric
explanation for the well-known ``polarization hole'' effect, in which the
sub-millimetre polarization percentage decreases toward the regions of peak
intensity. This occurs because of a cancellation between contributions to the
polarization from the ``backbone'' of poloidal flux along the filament's axis
and its surrounding envelope, which is dominated by the toroidal field
component. A systematic exploration of our parameter space allows us to
classify the polarization patterns due to filaments aligned approximately
perpendicular to the plane of the sky into three basic types. The polarization
vectors are parallel to filaments when $\ratio\appleq 0.1$, where $\Bzs$ and
$\Bphis$ are respectively the poloidal and toroidal magnetic field components
at the outer surface of the filament. The polarization vectors are
perpendicular to filaments when $\ratio\appgeq 0.33$. Intermediate cases result
in polarization patterns that contain $90^\circ$ flips in the orientation of
the polarization vectors. The flips are symmetric about the central axis for
filaments oriented parallel to the plane of the sky, but more complicated
asymmetric patters result from filaments that are inclined at some angle.
