We consider the problem of realizing abstract graph symmetries by symmetries of the 3-sphere through spatial embeddings. The techniques usually used in preceding studies are based on the placement of vertices and edges, and the problem has been solved for several well-known families of graphs. This paper provides a technique which is applicable to any graph, and gives an affirmative answer to the problem for abstract graph symmetries given by finite subgroups of O(4) with some property on their singular sets.

A 2n-component non-splittable spatial graph G = G1 ⋯ G2n in S3 possibly admits a smooth involution on its exterior whose fixed point set is a closed surface F of genus g ≥ 1. It is known that n ≥ g holds if G is a graph link in S3. In this paper, we consider G to be a hyperbolic spatial graph in a closed orientable 3-manifold M. We first study the case where M is the 3-sphere, and provide a method for constructing G and F with a1 + ⋯ + an > g > 1, where 1 - ai is the Euler characteristic of Gi for 1 ≤ i ≤ n. Next, we study the case where M is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries.

In this paper, we will show that for any link L in a closed orientable 3-manifold M, infinitely many hyperbolic 3-manifolds are obtained from M by Dehn surgeries so that each of them admits an orientation-preserving smooth finite cyclic group action with fixed point set L.

The aims of this paper is to prove that every closed connected orientable 3-manifold with an orientation-preserving periodic diffeomorphism contains infinitely many, setwise invariant, spatial graphs whose exteriors are hyperbolic 3-manifolds.

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

This paper provides a method for realizing an involution on an abstract graph, which does not restrict to an orientation-preserving automorphism on any cycle, by a rigidly achiral hyperbolic spatial embedding into the double of a compact connected orientable 3-manifold with non-empty aspherical boundary. As an application, we show that these hyperbolic spatial graphs are divided into two infinite families consisting of those which are homologically peripheral and those which are not.

If the exterior E(G) of a spatial graph G in a closed orientable 3-manifold is an irreducible 3-manifold with incompressible boundary, there is a unique finite graph Γ associated to the JSJ decomposition of E(G). This paper provides a method for constructing spatial graphs such that a given finite connected graph is associated to the JSJ decompositions of their exteriors. © 2013 World Scientific Publishing Company.

The Goeritz matrix of a connected link diagram is diagonalized to a normal form by the elementary divisor theory. The integer sequence given by the diagonal entries of this form is a link invariant called the Georitz invariant. In this paper, we study the Goeritz invariants of pretzel links. © 2013 Springer Basel.

Partially peripheral 3-manifolds and homologically peripheral 3-manifolds are defined by generalizing the notion of totally peripheral 3-manifolds. The aim of this paper is to provide a method for constructing hyperbolic links in 3-manifolds whose exteriors are not partially peripheral but homologically peripheral, and to prove that the connected sum with a homology 3-sphere enables every closed connected orientable 3-manifold to contain infinitely many such hyperbolic links.

A symmetry group of aspatial graph Gamma in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S-3 which leave Gamma setwise invariant. In this paper, we show that inmany cases symmetry groups of Gamma which agree on a regular neighborhood of Gamma are equivalent up to conjugate by rational twists along incompressible spheres and tori in thee xterior of Gamma

For any closed connected orientable 3-manifold M. we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S(3) via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M. (C) 2011 Elsevier B.V. All rights reserved.

We consider a graph link L in S(3) which admits a symmetry of the exterior given by a boundary component preserving periodic homeomorphism f (orientation-preserving or not). This paper shows that L is obtained by a specific construction method if f is not a strong inversion. As an application, we provide a complete classification of 2-component graph links with symmetries of the exteriors which are not extendable to symmetries of S(3).

dWe generalize the notion of totally peripheral 3-manifolds to define homologically peripheral 3-manifolds. The homologically peripheral property survives canonical decompositions of 3-manifolds as well as it defines a sufficiently large class of 3-manifolds containing link exteriors. The aim of this paper is to study finite group actions on a homologically peripheral 3-manifold, which agree on the boundary, up to equivalence relative to the boundary. As an application, we generalize Sakuma's theorems on the uniqueness of symmetries of knots to the case of symmetries of links.

We will classify 2-component non-splittable graph links in S(3), each of which is aspherically symmetric, i.e. the exterior admits a periodic homeomorphism which does not extend to a periodic homeomorphism of S(3). They can be obtained from spherically symmetric graph links by one or two steps of specific types of iterated cabling. Some of them admit both orientation-preserving and orientation-reversing aspherical symmetries. We will study whether they admit Dehn surgeries which yield 3-manifolds with the induced double symmetries or not.

A link L in S(3) possibly admits an involution of the exterior E(L) with fixed point set a closed surface, which is not extendable to an involution of S(3). In this paper, we focus on the case of graph links and show that the genus of the surface provides a lower estimate of the number of link components.

We will study boundaries of incompressible surfaces properly embedded in graph knot exteriors. We will first show that any bounded two-sided surface is meridional or longitudinal. In particular, iterated torus knot exteriors contain no bounded two-sided essential surface which is meridional or preferred longitudinal but Seifert surfaces. Even if the surface is possibly one-sided, the boundary is not of type (2p, 2q + 1) for any integers p and q.

An irreducible graph manifold M contains an essential torus if it is not a special Seifert manifold. Whether M contains a closed essential surface of negative Euler characteristic or not depends on the difference of Seifert fibrations from the two sides of a torus system which splits M into Seifert manifolds. However, it is not easy to characterize geometrically the class of irreducible graph manifolds which contain such surfaces. This article studies this problem in the case of graph link exteriors.

It has been studied much about PL finite group actions on 3-manifolds up to topological equivalence. The aim of this article is to study them in terms of PL equivalence. We first study the Hauptvermutung for polyhedral 3-orbifolds, and thereafter we prove topological equivalence implies PL equivalence.

Let ＄M＄ be a Haken manifold with a toral or empty boundary which is not a closed Seifert manifold. Zimmermann [31] showed that any mapping class ＄c＄ of ＄M＄ of finite order ＄n＄ can be realized by an auto-homeomorphism of period ＄n＄ using algebraic methods. In a special case, Heil and Tollefson [4], and Hong and McCullough [9] used methods from 3-manifold topology to approach the Zimmermann's theorem. This paper provides a complete proof by another topological approach.

Traffic measurement is an essential technique in moving toward the next stage of vehicular transportation, and one which needs to be more advanced in terms of both safety and efficiency. For this technique, image sensors are more advantageous than conventional sensors because of their portability and the wide variety of information they provide. Until now, however, vehicles overlapping each other within the image plane have caused problems with measurement accuracy. Therefore, a traffic measurement algorithm which features individual vehicle tracking using a rectangular model has been developed. The algorithm is used in real-time on the real-time vision system in order to utilize traffic volume to evaluate its performance. © 1996 IEEE.

We twist a knot by twisting some solid tori, each of which contains the knot, one by one to the meridional direction. Suppose the solid tori have mutually disjoint essential toral boundaries in the knot exterior. M. Kouno, K. Motegi and T. Shibuya have shown that, by twisting a knot along one or two solid tori in an inclusion relation, the resulting knot type is different from the original one (see [5] and [6]). In this paper, we investigate whether twisting along various solid tori also produces a different knot type or not. © 1993, International Academic Printing Co. Ltd., All rights reserved.

本研究課題は，３次元多様体上の有限群作用の研究をベースに、３次元多様体論で着目されているデーン手術等の議論を応用することにより，抽象グラフの対称性と３次元球面への空間埋め込みとの関係を調べることを目標としている。 まず，抽象グラフの対称性の空間埋め込みによる実現に関する研究を行った。抽象グラフの対称性が４次直交群の部分群で与えられるとき，３次元球面上の標準的線型作用のもとで集合として不変な空間埋め込みの存在について調べた。具体的に空間埋め込みを得るために，抽象グラフの対称性を容易に実現できる閉３次元多様体を構成し，同変デーン手術によって３次元球面へ変換する方法を確立した。そして，抽象グラフの底空間上の特異集合が３次元球面上の標準的線形作用の特異集合の部分集合に同相な場合に，肯定的に問題を解決した。研究成果は学術論文にまとめ，査読付き学術論文誌への掲載によって公表した。 次に，３次元多様体の対称性と枠つき絡み目の対称性との関係に関する研究を行った。３次元多様体上の向きを反転する滑らかな周期写像に対して，同変デーン手術により特異集合を縮小化すれば，３次元球面，２次元球面上の円周束，３次元トーラスのいずれかにおいて手術の記述が存在することを証明した。その過程では，先行研究で用いられた同変デーン手術による特異集合内の回転軸の変形操作に加え，曲面型の特異部分集合の圧縮を行う必要があり，これを同変デーン手術によって実現する方法を確立した。また、特異集合の縮小化以前の段階から手術の記述を得るためには，最大３段階の同変デーン手術が必要であることを示した。

(1) We showed that a link in the 3-sphere is the fixed point set of a cyclic group action on a 3-manifold obtained by Dehn surgery, and gave a condition for a spatial graph in the 3-sphere to have a symmetry given by an involution with fixed point set being a closed surface. (2) We proved that an orientation-reversing periodic diffeomorphism on a 3-manifold with a reduced fixed point set has a surgery description in either the 3-sphere, the circle-bundle over the 2-sphere, or the 3-torus. (3) We gave a condition for an abstract graph with a symmetry given by a finite subgroup of the orthogonal group O(4) to admit a spatial embedding which is setwise invariant under the linear action on the three-dimensional sphere.

Spaces are assumed to be simply connected and localized at a fixed odd prime p. We assume that the homology of each Hopf space has no p torsion, and the mod p cohomology ring of it is finite. We first completed the classification of irreducible Hopf spaces with rank less than or equal to five. Then, by using this result, we determined the irreducible decomposition of Hopf spaces satisfying the condition that the difference between the maximum and the minimum dimensions of the generators is less that 8(p-1). In the course of studying this problem we showed that some low dimensional unstable homotopy groups of spheres are generated by elements given by long Toda bracket of alpha elements. Moreover we extended a known fact on quasi p regularity of quaternionic Stiefel manifolds and real Stiefel manifolds.