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

粗略地说，纽结的能量是纽结空间上的一个泛函，它满足当纽结退化为具有双点的奇异纽结时爆炸的性质。在项目期间，我们与R。Langevin教授从积分几何的观点提出的泛函E与我们引入的泛函E之间的对应关系。因此，考虑到无穷小交比的概念，我们用能量的实部和绝对值来描述能量E，并得到一个描述朗之万引入量的公式。此外，我们从与无穷小交比和积分几何方法无关的讨论中新引入了几个共形不变泛函，除了上述观察之外，我们还注意到纽结的代数不变式的赋值与包含递归调用的程序的表示的赋值之间的强烈相似性，重点研究了程序语法中循环结构指称语义的一般机制。因此，我们得到了一个代数结构，使我们不仅能够模拟（无类型）λ-演算的计算范式，但功能体的内部结构。与一些已知结果相比，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，如跟踪monoidal范畴理论，找到一个整洁的关系，讨论能量的纽结。

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.