definition of the distance of a Heegaard surface generalizes to a notion of complexity for any knot that is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudoAnosov maps have minimal bridge presentations which are thin.

This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting. As a consequence, if a splitting surface has small genus compared to the translation distance of the monodromy, then the splitting is standard.

be closed, orientable 3-manifolds. Let H i denote a Heegaard surface in M i . We prove that if H 1 #H 2 comes from stabilizing a lower genus splitting of M 1 #M 2 then one of H 1 or H 2 comes from stabilizing a lower genus splitting. This answers a question of C Gordon [9, Problem 3.91]. We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of Heegaard splittings of prime 3-manifolds.

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.

The disk complex of a surface in a 3-manifold is used to define its topological index. Surfaces with well-defined topological index are shown to generalize well known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface H with topological index n to meet an incompressible surface F so that the sum of the indices of the components of H \ N(F) is at most n. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel's distance to surfaces with topological index ≥ 2.

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Abstract. In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a nonHaken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.

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