Abstract. Let M be a closed, irreducible, genus two 3–manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M. Then each component manifold Mi of M − F has handle number one, i.e. admits a Heegaard splitting obtained by attaching a single 1–handle to one or two components of ∂Mi. This result also holds for a decomposition of M along a maximal collection of incompressible tori. 1.

Abstract. We show that the only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Muñoz and by Morimoto-Sakuma. This confirms a conjecture first made by Goda and Teragaito. 1. Introduction and

ABSTRACT. Suppose N is a compressible boundary component of a compact irreducible orientable 3-manifold M and (Q,∂Q) ⊂ (M,∂M) is an orientable properly embedded essential surface in M in which some essential component is incident to N and no component is a disk. Let V and Q denote respectively the sets of vertices in the curve complex for N represented by boundaries of compressing disks and by boundary components of Q. Theorem: Suppose Q is essential in M, then d(V,Q) ≤ 1 − χ(Q). Hartshorn showed ([Ha]) that an incompressible surface in a closed 3-manifold puts a limit on the distance of any Heegaard splitting. An augmented version of the theorem above leads to a version of Hartshorn’s result for merely compact 3-manifolds. In a similar spirit, here is the main result: Theorem: Suppose a properly embedded connected surface Q is incident to N. Suppose further that Q is separating and compresses on both its sides, but not by way of disjoint disks. Then either • d(V,Q) ≤ 1 − χ(Q) or • Q is obtained from two nested connected incompressible boundaryparallel surfaces by a vertical tubing. Forthcoming work with M. Tomova ([STo]) will show how an augmented version of this theorem leads to the same conclusion as in Hartshorn’s theorem, not from an essential surface but from an alternate Heegaard surface. That is, if Q is a Heegaard splitting of a compact M then no other Heegaard splitting has distance greater than twice the genus of Q. 1.

Abstract. 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.) 1.

Abstract. We sketch a proof of the fact that a hyperbolic 3-manifold M with finitely generated fundamental group and with no parabolics are topologically tame. This proves the Marden’s conjecture. Our approach is to form an exhaustion Mi of M and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold Mi δ-hyperbolic and with Margulis constants independent of i. By taking the convex hull in the cover of Mi corresponding the core, we show that there exists an exiting sequence of surfaces Σi. We drill out the covers of Mi by a core C again to make it δ-hyperbolic. Then the boundary of the convex hull of Σi is shown to meet the core. By the compactness argument of Souto, we show that infinitely many of Σi are homotopic in M − C o.

Abstract. Suppose a link K in a 3-manifold M is in bridge position with respect to two different bridge surfaces P and Q, both of which are c-weakly incompressible in the complement of K. Then either • P and Q can be properly isotoped to intersect in a nonempty collection of curves that are essential on both surfaces, or • K is a Conway product with respect to an incompressible Conway sphere that naturally decomposes both P and Q into bridge surfaces for the respective factor link(s). 1.

In [Topology Appl. 90 (1998) 135] Scharleman showed that a strongly irreducible Heegaard splitting surface Q of a 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can

unified approach

The main result, Theorem 1, is Markov’s Theorem Without Stabilization (MTWS) for links in 3-space. Choose any oriented link type X and any closed braid representatives X−, X+ of X, where X+ has minimal braid index. The MTWS asserts that there is a complexity function and a finite set of ‘templates ’ such that (possibly after initial complexity-reducing modifications in the choices of X−, X+, which replace them with closed braids X ′ −, X ′ + of the same braid index) there is a sequence of closed braid representatives X ′ − = X 1 → X 2 → · · · → X r = X ′ + such that each passage X i → X i+1 is strictly complexity reducing and non-increasing on braid index. The templates which define the ‘moves ’ which take X i → X i+1 include 3 familiar ones: the destabilization, exchange move and admissible flype templates, and in addition, for each braid index m ≥ 4 a finite set T (m) of templates. The number of templates in T (m) is an increasing function of m for sufficiently large m. We give examples of members of T (m), but not a complete listing. There are consequences for the classification of transversal knots, i.e. knots which are everywhere transverse to the standard tight contact structure in S 3. Theorem 2 proves the existence

ABSTRACT. We study trivalent graphs in S 3 whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a level edge connecting two vertices. 1.