In this paper, we use normal surface theory to study Dehn filling on a knot-manifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knot-manifold that bound normal and almost normal surfaces in a one-vertex triangulation of that knot-manifold. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely which manifolds obtained by Dehn filling: 1) are reducible, 2) contain two-sided incompressible surfaces, 3) are Haken, 4) fiber over S 1 , 5) are the 3-sphere, and 6) are a lens space. Each of these algorithms is a finite computation.Moreover, in the case of essential surfaces, we show that the topology of the filled manifolds is strongly reflected in the triangulation of the knot-manifold. If a filled manifold contains an essential surface then the knot-manifold contains an essential vertex solution that caps off to an essential surface of the same type in the filled manifold. (Vertex solutions are the premier class of normal surface and are computable.) Algorithm I. Suppose X is a knot-manifold with a triangulation T which restricts to a one-vertex triangulation on ∂X. Given an embedded, twosided, closed, normal surface in (X, T ), determine precisely those slopes α for which the surface compresses in the Dehn filling X(α).Algorithm S. Given a knot-manifold X, determine precisely those slopes α for which the Dehn filling X(α) contains an embedded, incompressible, two-sided surface.We use Algorithm R and Algorithm S to give an algorithm to determine precisely those slopes for which the associated Dehn filling is a Hakenmanifold, Algorithm H.Algorithm H. Given a knot-manifold X, determine precisely those slopes α for which the Dehn filling X(α) is a Haken-manifold.At particular points in the application of Algorithm S, one may consider the alternative questions as to those slopes α for which the Dehn filling X(α) is either toroidal, the existence of an embedded, incompressible torus, or fibers over S 1 , the existence of an embedded, incompressible surface that is a fiber in such a fibration.Finally, in Section 6, we apply our techniques to similar considerations for Heegaard surfaces. We use almost normal surfaces introduced by H. Rubinstein [23] and thin position introduced by D. Gabai [4] as presented in the papers of Rubinstein [23] and A. Thompson [26]. The two main results of this section are given in Theorem 6.4 and Theorem 6.7 which provide algorithms to determine for a given knot-manifold X precisely those slopes α for which the Dehn filling X(α) is either S 3 or a lens space, respectively.Algorithm S. Given a knot-manifold X, determine precisely those slopes α for which X(α) is the 3-sphere.Algorithm L. Given a knot-manifold X, determine precisely those slopes α for which X(α) is a lens space.The authors wish to thank J. Hyam Rubinstein, whose collaborations with the first author have lead to many useful ideas and tools used in this
We show that if two 3-manifolds with toroidal boundary are glued via a "sufficiently complicated" map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, suppose X ∪ F Y is a manifold obtained by gluing X and Y , two connected small manifolds with incompressible boundary, along a closed surface F . Then the following inequality on genera is obtained:Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.
We demonstrate that for all but a finite number of Dehn fillings on a cusped manifold, the core of the attached solid torus is isotopic into every Heegaard surface for the filled manifold. Furthermore, if the cusped manifold does not contain a closed, non-peripheral, incompressible surface, then after excluding the aforementioned set and those filled manifolds containing incompressible surfaces (also a finite set) every other manifold obtained by Dehn filling contains at most a finite number of Heegaard surfaces that are not Heegaard surfaces for the cusped manifold. It follows that these manifolds contain a finite number of Heegaard surfaces of bounded genera. For each cusped manifold, the excluded manifolds are contained in a finite set that can be determined algorithmically.
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