The algorithm of finding planar surfaces in three-manifolds

By a slope in the boundary $\partial M$ of a 3-manifold, we mean the isotopy class $\alpha$ of a finite set of disjoint simple closed curves in $\partial M$ which are nontrivial and pairwise nonparallel. In this paper, we construct an algorithm to decide whether or not a given orientable 3-manifold $M$ contains an essential planar surface whose boundary has a given slope $\alpha$.