The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.     

Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

A method for constructing hyperbolic knots each of which bounds accidental incompressible Seifert surfaces of arbitrarily high genus is given. Mathematics Subject Classification (2000):57N10, 57M25.The author was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.     

Characterization of Totally Geodesic Totally Real 3-dimensional Submanifolds in the 6-sphere

In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.     

Totally Geodesic Invariant Subm anifolds of Contact Metric Manifold

§１．　ＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＭｂｅａ（２ｎ＋１）－ｄｉｍｅｎｓｉｏｎａｌｃｏｎｔａｃｔｍｅｔｒｉｃｍａｎｉｆｏｌｄｗｉｔｈｓｔｒｕｃｔｕｒｅｔｅｎｓｏｒｓ（Φ－，ξ－，η－，ｇ）．ＴｈｅｎｔｈｅｙｓａｔｉｓｆｙΦ－ξ－＝０，η－（ξ－）＝１，Φ－２＝－Ｉ＋η－ξ－，η－（Ｘ）＝ｇ（Ｘ，ξ－），　　　ｇ（Φ－Ｘ，Φ－Ｙ）＝ｇ（Ｘ，Ｙ）－η－（Ｘ）η－（Ｙ），ｇ（Ｘ，Φ－Ｙ）＝ｄη－（Ｘ，Ｙ）（１．１）ＦｏｒａｎｙｖｅｃｔｏｒｆｉｅｌｄｓＸａｎｄＹ…     

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．     

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.

     

纽结补中的不可压缩分段不可压缩曲面

本文利用扭转交叉数（twist-erossing number)讨论扭结补中的不可压缩分段不可压缩曲面的性质,设K是一个排叉结（pretzel knot)或者是一个扭转交叉数少于6的有理纽结,如果F是S^3-K中的不可压缩分段不可压缩曲面,那么F是一个穿孔球面。     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L^6g+2b+2c–6. This answers a long-standing open question.     

Real Places and Torus Bundles

If M is a hyperbolic once-punctured torus bundle over S ¹, then the trace field of M has no real places.     

The Compressibility of Checkerboard Surfaces of Link Diagrams

Consider the checkerboard surfaces defined by some link diagrams. When they are not orientable, one considers the boundary surfaces of small regular neighborhoods of them. This article studies the compressibility problem of these kinds of surfaces in the link complements. The problem is solved by devising a normalization theory for the compressing discs, which brings up an algorithm to read out compressibility directly from the link diagrams. As an application of the algorithm, the compressibility changes under Reidermeister moves are studied. Diagrams from the knot tables are also studied, and surprisingly, some of them are shown to define completely compressible surfaces of this kind. Infinitely many examples of non-alternating knot diagrams with incompressible surfaces of this kind are also constructed.     

A Kind of Essential Surfaces in the Complements of Knots

In this paper, by the twist-crossing number of knots, we give an upper bound on the Euler characteristic of a kind of essential surfaces in the complements of alternating knots and almost alternating knots, which improves the estimation of the Euler characteristic of the essential surfaces with boundaries under certain conditions. Furthermore, we give the genus of the essential surfaces.     

The Distribution of Geodesic Excursions Out the End of a Hyperbolic Orbifold and Approximation with Respect to a Fuchsian Group

A generic geodesic on a finite area, noncompact surface with a hyperbolic orbifold structure travels densely about the surface, making excursions out the non-compact end and then returning to a compact piece of the surface. We show that for almost all geodesics on such a surface there is a limiting distribution for the sequence of depths of the excursions out a noncompact end. The distribution is independent of the surface and closely resembles the distribution arising from a sequence of continued fraction approximations in the metrical theory of diophantine approximation. This metrical theory can then be reproduced in the setting of approximation by parabolic fixed points in a Fuchsian group.     

简单拓扑图及几乎交错链环补中的闭曲面*

研究了在曲面拓扑图的每个球泡中最多有一个鞍点的情况下,拓扑图的简单性与曲面亏格的关系. 并利用正方形素方割的性质讨论了素、非分离几乎交错链环补空间中的闭、不可压缩、分段不可压缩曲 面的性质.     

二维复空间型的全实曲面

本文有兴趣于二维复空间型的全实曲面的研究．得到了一些有趣结果，并将［１，２］中的有关定理推广到具平行平均曲率向量的全实曲面的情形．     

The Canonical Decompositions of Some Family of Compact Orientable Hyperbolic 3-Manifolds with Totally Geodesic Boundary

L. Paoluzzi and B. Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism. Our main purpose is to determine the canonical decompositions of these manifolds. Using the result, we can obtain an alternative proof of the classification theorem of these manifolds and determine their isometry groups. We also determine their unknotting tunnels. Some of these manifolds are related to certain spatial graphs, so-called Suzukis Brunnian graphs. The properties of these manifolds enable us to obtain those of the graphs. Moreover, we give an affirmative answer to Kinoshitas problem concerning these graphs. In the Appendix, we calculate the volume of these manifolds.     

Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

Caylay射影平面的全实曲面

It has been shown, under certain conditions on the Gauss curvature, every totally real surface of the Cayley projective plane with parallel mean curvature vector is either flat or totally geodesic.     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.