Research on various geometric structures on 3-manifolds

3-流形上的各种几何结构研究

• 批准号: 15540073
• 负责人: KOBAYASHI Tsuyoshi
• 金额: $ 1.34万
• 依托单位: Nara Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540073/
• 关键词: 3-manifolds; Heegaard splittings; knots; tunnel number; Haken manifold; automatic group; virtual Haken予想

1.Morimoto's Conjecture on the tunnel numbers of composite knots in 3-manifoldsLet t(K) be the tunnel number of a knot K in a 3-manifold. Suppose for m-small knots K_1,…,K_n, the super additivity of tunnel number does not hold for #^n_<i=1> K_i, Then we proved that there exists a subset I of {1,【triple bond】,n} such that #_<i∈1>/K_i admits a primitive meridian.2.The growth rate of tunnel number of knotsFor a knot K in a 3-manifold M, we defined a numerical invariant called the growth rate of the tunnel numbers of K, and proved the following.Suppose that the Heegaard genus of K is greater than the Heegaard genus of M. Then the growth rate of the tunnel numbers of K is less than 1.3.Gersten's Problem for automatic groupGersten posed the following problem "Each automatic group is eithr (1)a finite group, (2)contains free abelian group of rank 2,or (3)a word hyperbolic group." We showed that for a class of automatic group (called n-starred groups) this problem is solved affirmatively.4.Heegaard gradients Seifert fibered spacesWe completely determined for which Seifert fibered space, the Heegaard gradient vanish.

1。莫里莫托（Morimoto）在3个manifolds中复合结的隧道数量上的猜想，使t（k）是3个manifold中的结k的隧道数。假设对于m-small结k_1，…，k_n，隧道编号的超级添加性不适用于＃^n_ <i = 1> k_i，然后我们证明存在一个子集I {1，[triple bond]，n}，因此＃_ <i∈1>/k_i允许原始的子午线。定义的一个数值不变式称为K的隧道数量的增长率，并证明了以下几个数字。plapsepspopplands，k的Heegaard属大于M的Heegaard属。那么K的隧道数量的增长速率小于1.3.Gersten的问题。ggersten的问题是自动组成的小组，每个自动组都在2个自动组中，均为一个（1）contiite contient contematian conte contematian contematian conte conte a a co. （3）单词双曲线群。”我们表明，对于一类自动组（称为n星组），此问题是积极解决的。4.heegaard梯度Seifert Fibiber的SpaceSwe完全确定了Seifert Fiber Space，Heegaard梯度消失了。

项目成果

Kazuhiro Ichihara, Masakazu Teragaito: "Klein bottle surgery and genera of knots"Pacific J.Math.. 210(2). 317-333 (2003)

Kazuhiro Ichihara、Masakazu Teragaito：“克莱因瓶手术和结属”Pacific J.Math.. 210(2)。

Locally thin position for a link

链接的局部薄位置

• 发表时间: 2003
• 期刊: Kobe J.Math. 20
• 作者: Kobayashi;Teiichi;Minyou Katagiri;T.Shibata;Tsuyoshi Kobayashi;T.Kobayashi;Tsuyoshi Kobayashi;S.Matsumoto;T.Inaba;Kazuhiro Ichihara;Tsuyoshi Kobayashi
• 通讯作者: Tsuyoshi Kobayashi

On conformally flat critical Riemannian metrics for a curvature functional

• DOI: 10.3792/pjaa.81.27
• 发表时间: 2005-02
• 作者: M. Katagiri

Heegaard gradient of Seifert fibered 3-manifolds

Seifert 纤维 3 流形的 Heegaard 梯度

• 发表时间: 2004
• 期刊: Bulletin of the London Mathematical Society 36
• 作者: Tomohisa Inoue;Juno Mukai;Kazuhiro Ichihara
• 通讯作者: Kazuhiro Ichihara

On the growth rate of tunnel number of knots

论隧道节数增长率

• 期刊: Journal fur die Reine and Ang. Math. (to appear)
• 作者: Kobayashi;Teiichi;Minyou Katagiri;T.Shibata;Tsuyoshi Kobayashi;T.Kobayashi;Tsuyoshi Kobayashi;S.Matsumoto;T.Inaba;Kazuhiro Ichihara;Tsuyoshi Kobayashi;Kazuhiro Ichihara;Tsuyoshi Kobayashi
• 通讯作者: Tsuyoshi Kobayashi