Exceptional Dehn surgery creating essential or Heegaard tori

出色的 Dehn 手术创造了必要的或 Heegaard 环面

• 批准号: 16540071
• 负责人: TERAGAITO Masakazu
• 金额: $ 1.34万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540071/
• 关键词: knot; 3-manifold; Dehn surgery; 双曲3次元多様体; 国際情報交換; 韓国

In this research, we focused on exceptional Dehn surgery and filling for hyperbolic 3-manifolds which create essential tori or Heegaard tori. Precisely, we studied the following problems and obtained the results.(1) Manifolds realizing the maximal distance between toroidal Dehn surgeries: First, we determined all knots that admit two toroidal Dehn surgeries at distance 5. Second, we showed that the distance between toroidal Dehn fillings on a large hyperbolic 3-manifold is at most 4. Finally, we proved that if a hyperbolic manifold admits a toroidal Dehn filling and an annular Dehn filling at distance 3, then the boundary of the manifold consists of at most two tori.(2) Sequence of integral exceptional Dehn surgeries: Except Eudave-Munoz knots, it is conjectured that any exceptional Dehn surgery is integral. We studied all known examples of exceptional Dehn surgery, and found that integral exceptional Dehn surgeries form consecutive integers. By using the pentangle, we constructed hyperbolic knots which admit multiple exceptional Dehn surgeries.(3) Non-integral Dehn surgeries creating closed non-orientable surfaces : We showed that any closed non-orientable surface of genus greater than two can be created by non-integral Dehn surgery on hyperbolic knots.(4) Alexander polynomials of doubly primitive knots: We gave a formula of Alexander polynomial of doubly primitive knots. As a consequence, we showed that the Alexander polynomial of a doubly primitive knot has +1 and-1 alternatively as its coefficient.(5) A Seifert fibered manifold with infinitely many knot-surgery descriptions: We gave the first example of a small Seifert fibered manifold that can be obtained from infinitely many hyperbolic knots by the same Dehn surgery. In particular, our knots have no symmetry, so that they cannot lie on a genus two Heegaard surface of the 3-sphere.

在这项研究中，我们专注于特殊的Dehn手术和填充双曲3-流形，创造必要的环面或Heegaard环面。准确地说，我们研究了以下问题并得到了结果。(1)实现环形Dehn手术之间的最大距离的流形：首先，我们确定了在距离5处允许两个环形Dehn手术的所有结。其次，我们证明了在一个大的双曲三维流形上，环形Dehn填充之间的距离至多为4。最后，我们证明了：如果一个双曲流形在距离3处允许一个环面Dehn填充和一个环形Dehn填充，则该流形的边界至多由两个环面组成。(2)完整的特殊Dehn手术顺序：除Eudave-Munoz结外，任何特殊Dehn手术都是完整的。我们研究了所有已知的例外Dehn手术的例子，发现整数例外Dehn手术形成连续整数。通过使用五角形，我们构造了双曲结，允许多个特殊的Dehn手术。(3)非积分Dehn手术创建封闭的不可定向曲面：我们证明了亏格大于2的任何封闭的不可定向曲面都可以通过双曲结上的非积分Dehn手术创建。(4)双本原纽结的亚历山大多项式：给出了双本原纽结的亚历山大多项式的一个公式。结果表明，双本原纽结的亚历山大多项式的系数是+1和-1交替的。(5)一个具有无穷多个knot-surgery描述的塞弗特纤维流形：我们给出了一个小塞弗特纤维流形的第一个例子，它可以通过相同的Dehn手术从无穷多个双曲纽结中获得。特别是，我们的纽结没有对称性，因此它们不能位于3-球面的亏格为2的Heegaard曲面上。

项目成果

Toroidal Dehn fillings on large hyperbolic 3-manifolds

大型双曲 3 流形上的环形 Dehn 填充物

• 发表时间: 2006
• 期刊: Comm.. Anal. Geom. 14
• 作者: Hiroyuki Ito;Kimura Shunichi;Masakazu Teragaito
• 通讯作者: Masakazu Teragaito

Boundary structure of hyperbolic 3-manifolds admitting annular and toroidal fillings at large distance

允许大距离环形和环形填充的双曲3流形的边界结构

• 期刊: Canadian Journal of Mathematics (印刷中)
• 作者: Hiroshi Goda;Masakazu Teragaito;Masakazu Teragaito
• 通讯作者: Masakazu Teragaito

Alexander polynomials of doubly primitive knots

双原结的亚历山大多项式

• 发表时间: 2007
• 期刊: Proceedings of the American Mathematical Society 135・2
• 作者: Nakajima;Hiraku;Kenji Fukaya;Akira Kono;Akira Kono;Hiraku Nakajima;Hiraku Nakajima;Masakazu Teragaito
• 通讯作者: Masakazu Teragaito

On hyperbolic knots realizing the maximal distance between toroidal Dehn surgeries

双曲线结实现环形Dehn手术之间的最大距离

• 发表时间: 2006
• 期刊: Journal of Knot Theory and its Ramifications 15・1
• 作者: Kazuhiro Ichihara;Toshio Saito;Masakazu Teragaito;Masakazu Teragaito
• 通讯作者: Masakazu Teragaito

On hyperbolic knots realizing the maximal distance between toroidal surgeries

实现环形手术间最大距离的双曲结

• 发表时间: 2006
• 期刊: Journal of Knot Theory and its Ramifications 15, no.1
• 作者: Kazuhiro Ichihara;Toshio Saito;Masakazu Teragaito;Masakazu Teragaito;Masakazu Teragaito;Masakazu Teragaito;Masakazu Teragaito
• 通讯作者: Masakazu Teragaito