Research of topological invariants of plane closed curves

本研究的主要目的是，首先得到闭平面曲线的新的拓扑不变量，然后研究它们的几何和代数性质。首先，我们得到了两个无穷级数的拓扑不变量，它们分别表示为i^&lt；epsilon&gt；_&lt；habeta&gt；和ST_k，其中epsilon为+，0或-，并且α，β，k在所有自然数的集合上变化。本研究的一个重要结果是证明了不变量i^&lt；epsilon&gt；_&lt；habeta&gt；证明了这些不变量之间存在无穷多个代数独立的拓扑不变量；对于不变量ST_k，研究了它们在不稳定曲线上沿规则变形的跳跃，验证了与V.I.Arnold得到的奇异性不变量相同的几何性质，以及它们之间的代数无关性和关于平面曲线连通和运算的可加性.我们还得到了解释i^-lt；epsilon-gt；--lt；habeta&gt；与ST_k之间关系的公式。

The mail purposes of this research are, firstly to obtain new topological invariants for closed plane curves, and secondly to investigate their geometric and algebraic properties.For the first purpose, we obtained two infinite series of topological invariants which are denoted by I^<epsilon>_<habeta> and St_k, where epsilon is +, 0 or -, and alpha, beta, and k vary over the set of all natural numbers.One of the important results of this research was to show the order in the sense of Vassiliev of the invariants I^<epsilon>_<habeta> to be equal to alpha + 1. Establishing the independence among those invariants, we have shown that there exist, for all finite order, infinitely many algebraically independent topological invariants.For the invariants St_k, investigating the jumps of their values at the unstable curves along regular deformations, we have verified the same geometric property as the strangeness invariants obtained by V.I.Arnold, and also algebraic independence among them as well as the additivity with respect to the connected sum operation of plane curves. We have also obtained a formula explaining the relation between I^<epsilon>_<habeta> and St_k.