m at h . G T ] 2 8 A ug 2 00 6 FINDING PLANAR SURFACES IN KNOT - AND LINK - MANIFOLDS

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. The method uses normal surface theory but does not follow the classical approach. Here the proof uses a rewriting method for normal surfaces in a fixed triangulation and may not find the desired solution among the fundamental surfaces. Two major results are obtained under certain boundary conditions. Given a link-manifold M , a component B of ∂M , and a slope γ on B, it is shown that there is an algorithm to decide if there is an embedded punctured-disk in M with boundary γ and punctures in ∂M \ B; if there is one, the algorithm will construct one. Again, while normal surfaces are used, we may not find a solution among the fundamental surfaces. In this case we use induction on the number of boundary components of the link-manifold. It also is shown that given a link-manifold M , a component B of ∂M , and a meridian slope µ on B, there is an algorithm to decide if there is an embedded punctured-disk with boundary a longitude on B and punctures in ∂M \ B; if there is one, the algorithm will construct one. This is shown to follow from the previous result using a link-manifold related to M and called the link-manifold obtained from M by Dehn drilling along the slope µ. The 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; the average length estimate shows, in quite general situations, that given the link-manifold M by a triangulation T , then all normal surfaces of a bounded genus must have a short boundary curve on some boundary of M. The constant that determines how short is completely determined by the fundamental surfaces in (M, T). A version of the average length estimate with boundary conditions also is derived.