  It is shown that given any link-manifold, there is an algorithm to decide if
the manifold contains an embedded, essential planar surface; if it does, the
algorithm will construct one. If a slope on the boundary of the link-manifold
is given, there is an algorithm to determine if the slope bounds an embedded
punctured-disk; if a meridian slope is given, then it can be determined if a
longitude bounds an embedded punctured-disk. The methods use normal surface
theory but do not follow the classical approach. Properties of minimal vertex
triangulations, layered-triangulations, 0--efficient triangulations and
especially triangulated Dehn fillings are central to our methods. We also use
an average length estimate for boundary curves of embedded normal surfaces; a
version of the average length estimate with boundary conditions also is
derived. An algorithm is given to construct precisely those slopes on the
boundary of a given link-manifold that bound an embedded punctured-disk in the
link-manifold.
