Incompressible surfaces in handlebodies and closed 3-manifolds of Heegaard genus 2

In this paper, we shall prove that for any integer 0$">, 1) a handlebody of genus 2 contains a separating incompressible surface of genus , 2) there exists a closed 3-manifold of Heegaard genus which contains a separating incompressible surface of genus .

     

Incompressible surfaces in handlebodies and boundary reducible 3-manifolds

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.     

Totally knotted Seifert surfaces with accidental peripherals

We show that if there exists an essential accidental surface in the knot exterior, then a closed accidental surface also exists. As its corollary, we know boundary slopes of accidental essential surfaces are integral or meridional. It is shown that an accidental incompressible Seifert surface in knot exteriors in is totally knotted. Examples of satellite knots with arbitrarily high genus Seifert surfaces with accidental peripherals are given, and a Haken 3-manifold which contains a hyperbolic knot with an accidental incompressible Seifert surface of genus one is also given.

     

On the mapping class group of a genus 2 handlebody

A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody.     

Critical Heegaard surfaces

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 non-Haken 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.     

几乎交错环链补中的不可压缩曲面

In this paper,we discuss mainly the properties of incompressible pairwies incom- prcssiblc surfaccs in almost altcrnating link complcmcnts. Lct L bca almost link and lct F be an incompressible palrwise 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.     

Incompressible pairwise incompressible surfaces in almost alternating knot complements

In this paper, we deal with incompressible pairwise incompressible surfaces in almost alternating knot complements. We show that the genus of a surface in an almost alternating knot exterior equals zero if there are two, four or six boundary components in the surface.     

平环和的Heegaard亏格的一个下界

Let Mi, i = 1,2, be a compact orientable 3-manifold, and Ai an incompressible annulus on a component Fi of OMi. Suppose A1 is separating on F1 and A2 is non-separating on F2. Let M be the annulus sum of M1 and M2 along A1 and A2. In the present paper, we give a lower bound for the genus of the annulus sum M in the condition of the Heegaard distances of the submanifolds M1 and M2     

几乎交错纽结补中决定的不可压缩曲面

本文讨论了几乎交错纽结补中的不压缩，两两不可压缩曲面的性质．证明了，当Ｋ是素的几乎交错纽结时，若ＦＳ３－Ｋ是不可压缩、两两不可压缩曲面，则对于固定的边界分支数ｎ，曲面Ｆ的合痕类是有限的．     

A complex of incompressible surfaces for handlebodies and the mapping class group

For a genus g handlebody H _g a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H _g , is constructed and several properties are established. In particular, this complex naturally contains, as a subcomplex, the complex of curves of the surface ${\partial H_{g}}$ . As in the classical theory, the group of automorphisms of this complex is identified with the mapping class group of the handlebody.     

On upper bounds on stable commutator lengths in mapping class groups

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.     

Homologically versus homotopically area minimizing surfaces in 3-manifolds

We construct a Riemannian metric on the 3-torus such that no closed surface minimizing area in its homology class is incompressible, i.e., each such surface is of genus greater than one. In particular, for such a Riemannian metric, the homotopically area minimizing 2-tori constructed in [5] do not minimize area in their homology classes. The example is easily generalized to arbitrary 3-manifolds. The constructed Riemannian metric can be chosen to be conformally equivalent to any arbitrary given one. Received September 4, 1998 / Accepted October 23, 1998     

Bilateral self-amalgamation of a Heegaard splitting and Hempel distance

A closed orientable Haken 3-manifold containing a non separating incompressible closed surface has two canonical Heegaard splittings, which are called self-amalgamation and bilateral self-amalgamation.Heegaard distance introduced by Hempel is a useful index in studying Heegaard splitting. This paper studies the stabilization problem for the bilateral self-amalgamation, and proves that if the distance of bilateral selfamalgamation of a Heegaard splitting is at least 9, then it is unstabilized, weakly reducible and irreducible.     

A cut finite element method for a Stokes interface problem

We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.     

A stall-delay model for rotating blades

The paper is aimed at describing a fundamental phenomenon: the effect of rotation on the inboard blade boundary layer on a wind turbine. The three-dimensional incompressible steady momentum integral boundary-layer equations are used to predict the boundary-layer growth and limiting streamline angles on the blade surface for both attached and separating flow. The chordwise skin friction coefficient is used to identify boundary layer separation and shear layer reattachment locations. The nature of flow near the axis of rotation is discussed and the physical mechanism associated with 3-D and rotational effects is identified. A semi-empirical correction law for the lift coefficient based on 2-D airfoil data is established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node

In this article, we consider a family of compact Riemann surfaces of genus q 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.     

Hirzebruch χy genera of the Hilbert schemes of surfaces by localization formula

We use the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula to calculate the Hirzebruch χy genus χy(S[n]), where S[n] is the Hilbert scheme of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on (C2)[n] is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes.     

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.     

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 S~3 - L. We discuss the properties that the surface F intersects with 2-spheres in S~3 - 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 S~2-move), and define the characteristic number of the topological graph for F∩S~2±. 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∩S~2+(or F∩S~2-) 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.