  When a Dehn filled link manifold contains a geometrically incompressible
one-sided surface, it is shown there is a unique boundary incompressible
position that the surface can take in the link space. The proof uses a version
of the sweep-out technique from two-sided Heegaard splitting theory. When
applied to one-sided Heegaard splittings, this result can be used to complete
the classification of one-sided splittings of (2p, q) fillings of Figure 8 knot
space: determining that fillings with |2p/q|<3 have two non-isotopic
geometrically incompressible one-sided splitting surfaces.
