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Energy of knots (existence of energy minimizers and numerical experiment)

结的能量（能量最小化器的存在性和数值实验）

基本信息

批准号：
10640085
负责人：
KURATA Toshihiko
金额：
$ 1.79万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640085/
关键词：
topology knot energy 低次元トポロジー結び目のエネルギー

项目摘要

Roughly speaking, energy of knots is a functional on the space of knots which fulfills the property that it explodes whenever a knot degenerates to a singular knot with double points. During the period of the project, we have studied this concept of energy functional together with Prof. R. Langevin of Universit de Bourgogne mainly to clarify the correspondence between our introduced functional E and the one due to Prof. Langevin which is contrived from the viewpoint of integral geometry. As a result, considering the notion of infinitesimal cross-ratio, we described the energy E in terms of its real-part and its absolute value, and also obtained a formula which describes Langevin's introduced quantity. Furthermore, we newly introduced several conformally invariant functional from a discussion independent of the approach of infinitesimal cross-ratio and integral geometry.Besides the observation above, we also direct our attention to a strong analogy between the assignment of algebraic invariant of knots and the assignment of denotation of programs containing recursive-calls, and especially study the general mechanism of denotational semantics of cyclic structures appearing in the syntax of programs. In consequence, we obtained an algebraic structure which enables us to model not only the computational paradigm of (type-free) λ-calculus but the internal structure of functional body. Our construction can be regarded as enough simple in comparison with some known results, and so we expect to find a neat relationship to the discussion of energy of knots through a general framework, such as the theory of traced monoidal categories.

粗略地说，结点的能量是结点空间上的一个泛函，它满足了当一个结点退化为具有双点的奇异结点时它爆炸的性质。在项目期间，我们与勃艮第大学R. Langevin教授一起研究了这个能量泛函的概念，主要是为了澄清我们引入的泛函E与由Langevin教授从积分几何的角度提出的E之间的对应关系。因此，考虑到极小交叉比的概念，我们用能量E的实部和绝对值来描述能量E，并得到了一个描述朗之万引入量的公式。此外，我们从独立于无穷小交叉比和积分几何的讨论中引入了几个保形不变泛函。除上述观察外，我们还注意到结点的代数不变量的赋值与包含递归调用的程序的外延赋值之间的强烈类比，并特别研究了循环结构在程序语法中出现的外延语义的一般机制。因此，我们得到了一个代数结构，使我们不仅可以模拟（无类型）λ-微积分的计算范式，而且可以模拟功能体的内部结构。与一些已知的结果相比，我们的构造可以被认为是足够简单的，因此我们期望通过一个一般的框架，如迹一元范畴理论，找到一个与节能量讨论的简洁的关系。