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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

基本信息

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

项目摘要

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。除了光滑的2-结之外，解结的蚂蚁条件是众所周知的。补有相同的同伦类型与平凡结的条件。我们的目的是证明这个条件也是足够的剩余的情况下，使用2维辫子理论的方法是相当初级的，我们希望找到一些数学的本质在这个机会。1）我们已经有一个单参数的家庭的地图只有尖点出生和死亡之间的一个给定的结的条件和平凡的结。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，通过应用稳定化找到其图表描述。4)交叉点的高度可以向下修改，直到水平略高于相关的尖灭。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)这个步骤可以通过Kamada-Matsumoto的词表示法分为几个简单的步骤。7)每一步都将表明，末端是平凡的二维编织物。这是解决猜想的大纲。至于数学的本质，除了每位研究者的研究之外，我们还研究了关高和去世300年后的行列式历史。