Incompressible Pairwise Incompressible Surfaces in Knot Exteriors

We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.

INCOMPRESSIBLE PAIRWISE INCOMPRESSIBLE SURFACES IN LINK COMPLEMENTS

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S3 - L. We discuss the properties that the surface F intersects with 2-spheres in S3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves （R-move and S2-move）, and define the characteristic number of the topological graph for F∩S2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating （or almost alternating） link complements. We prove that the genus of the surface equals zero if the component number of F∩S2＋（or F∩S2-） is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #（F） ≤8.

Incompressible Surfaces in Almost Alternating Link Exteriors

In this paper, we discuss mainly the properties of incompressible pairwise incomprcssiblc surfaccs in ahnost altcrnating link complcmcnts. Lct L bc a almost link and lct F be an incompressible pairwise incompressible surface in S^3 - L. First, we give the properties that the surface F intersects with 2-spheres in S^3- L. The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph. One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph. Next, we prove that if the graph is special simple, then the genus of the surface is zero.

The Existence of a Meridional Curve in Closed Incompressible Surfaces in Fully Alternating Link Complements

Menasco showed that a closed incompressible surface in the complement of a non-split prime alternating link in S^(3) contains a circle isotopic in the link complement to a meridian of the links.Based on this result,he was able to argue the hyperbolicity of non-split prime alternating links in S3.Adams et al.showed that if F■S×I\L is an essential torus,then F contains a circle which is isotopic in S×I\L to a meridian of L.The author generalizes his result as follows:Let S be a closed orientable surface,L be a fully alternating link in S×I\If F ■ S×I\L is a closed essential surface,then F contains a circle which is isotopic in S×I\L to a meridian of L.

A standard form of incompressible surfaces in 3-dimensional handlebodies

Applying the Morse theory,we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies.We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form.Our algorithm is practical.Several examples are given to test the algorithm.

3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera

In this paper, we shall prove that for any orientable 3-manifold M, there is a link L - K U K1 U K2 U K3 with four components in M, such that the complement of L, say ML, contains separating essential closed surfaces of all positive genera.

Reducing a Representation of Homotopy 3-spheres

In the present note we show that a representation of homotopy 3-spheres can be somewhat simplified under some circumstances.

Haken 3-Manifolds in Small Covers

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope.In addition,they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.