Smooth unknotting conjecture in dimension four and search for essence of mathematics

四维光滑解结猜想探寻数学本质

• 批准号: 16340017
• 负责人: MATUMOTO Takao
• 金额: $ 11.12万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340017/
• 关键词: Topologv; Knot; Braid; Chart description; Cusp; 1-parameter family; Elementary method; Historical search; 2次元結び目; 2次元ブレイド; マルコフ型定理; チャート表示の変形; 解け予想; 初等的手法; 変形; カンドル; 初等的; 2次元滑らか結び目; 結び目解け予想; 安定化

The necessary and. ant condition for unknotting is well-known except for a smooth 2-knot. The complement has the same homotopy type with a trivial knot is the condition. Our aim is to show that this condition is enough also for the remaining case, using 2-dimensional braid theory The method is quite elementary and we want to find some essence of mathematics in this chance.1) We have already a one-parameter family of maps with only cusp births and deaths between a given knot with the condition and a trivial knot. 2) Transform this to a one-parameter family of singular 2-braids by extending a method due to Kamada. 3) Find its chart description by applying stabilization if necessary. 4) The height of an intersection point can be modified down until the level a little higher than the related cusp death. 5) This reduces the problem to the case that the number of intersection points is one and the intersection point go down by rotating the other fixed vertices of the chart and then disappear at the cusp. 6) This step can be divided into several simple steps by a word representation due to Kamada-Matsumoto. 7) Each step will be shown that the ends are turned out to be trivial 2-dimensional braids. This is an outline to solve the conjecture.As for essence of mathematics besides the study by each investigator we studied the history of determinants at the occasion of 300 years after the death of Takakazu Seki.

必要的和。除平滑的 2 结外，解结的蚂蚁条件是众所周知的。补体具有相同的同伦类型，并且有平凡结是条件。我们的目的是使用二维辫子理论证明这个条件对于其余情况也足够。该方法非常基本，我们希望借此机会找到一些数学本质。1）我们已经有了一个单参数族具有条件的给定结和平凡结之间仅具有尖点出生和死亡的地图。 2) 通过扩展 Kamada 的方法，将其转换为奇异 2 辫子的单参数族。 3) 如有必要，通过应用稳定来查找其图表描述。 4) 可以向下修改交点的高度，直到略高于相关尖点死亡的水平。 5）这将问题简化为交点的数量为1并且交点通过旋转图表的其他固定顶点而下降，然后在尖点处消失的情况。 6) 由于 Kamada-Matsumoto，该步骤可以通过单词表示分为几个简单的步骤。 7) 每一步都会显示末端是平凡的二维辫子。这是解决猜想的大纲。 至于数学的本质，除了各位研究者的研究之外，我们还研究了关贵一死后300年的行列式的历史。

项目成果

On the smooth unknotting conjecture in dimension four

关于第四维光滑解结猜想

• 发表时间: 2008
• 作者: 松本 堯生

Braid presentation of virtual knots and welded knots

虚拟结和焊接结的编织演示

• 发表时间: 2007
• 作者: Seiichi Kamada

Word representation of cords on a punctured plane

刺穿平面上绳索的文字表示

• 发表时间: 2005
• 作者: Seiichi; Kameda; Yukio; Matsumoto
• 通讯作者: Matsumoto

2次元ブレイドと4次元結び目

2D 编织和 4D 结

• 发表时间: 2005
• 作者: 鎌田聖一

on the smooth unknotting comjecture in dimension four

关于第四维光滑解结猜想

• 发表时间: 2008
• 作者: Takao; Matuinoto
• 通讯作者: Matuinoto