2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115594It 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.44 pages, 9 figuresGeometric TopologyPrimary 57N10, 57M99; Secondary 57M50text