Heegaard Splittings with Boundary and Almost Normal Surfaces
نویسنده
چکیده
This paper generalizes the definition of a Heegaard splitting to unify the concepts of thin position for 3-manifolds [15], thin position for knots [2], and normal and almost normal surface theory [3], [13]. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In the final section, we use this machinery to produce an algorithm to determine the bridge number of a knot, provided thin position for the knot coincides with bridge position. We also present several algorithmic and finiteness results about Dehn fillings with small
منابع مشابه
Normal Surfaces
We define a 2-normal surface to be one which intersects every 3-simplex of a triangulated 3-manifold in normal triangles and quadrilaterals, with one or two exceptions. The possible exceptions are a pair of octagons, a pair of unknotted tubes, an octagon and a tube, or a 12-gon. In this paper we use the theory of critical surfaces developed in [Baca] to prove the existence of topologically inte...
متن کاملLayered - Triangulations of 3 – Manifolds
A family of one-vertex triangulations of the genus-g-handlebody, called layered-triangulations, is defined. These triangulations induce a one-vertex triangulation on the boundary of the handlebody, a genus g surface. Conversely, any one-vertex triangulation of a genus g surface can be placed on the boundary of the genus-g-handlebody in infinitely many distinct ways; it is shown that any of thes...
متن کاملHeegaard Surfaces and the Distance of Amalgamation
Let M1 and M2 be orientable irreducible 3–manifolds with connected boundary and suppose ∂M1 = ∂M2. Let M be a closed 3–manifold obtained by gluing M1 to M2 along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then M is not homeomorphic to S and all small-genus Heegaard splittings of M are standard in a certain sense. In particular, g(M) = g(M1) + g(M2) − g(∂...
متن کاملHeegaard Splittings of Sufficiently Complicated 3-manifolds I: Stabilization
We construct families of pairs of Heegaard splittings that must be stabilized several times to become equivalent. The first such pair differs only by their orientation. These are genus n splittings of a closed 3-manifold that must be stabilized at least n − 2 times to become equivalent. The second is a pair of genus n splittings of a manifold with toroidal boundary that must be stabilized at le...
متن کاملOn Non-compact Heegaard Splittings
A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main theorem is: if N is a compact, connected, orientable 3-manifold with non-empty boundary, with no S components, and if M is obtained from N by...
متن کامل