STUDY ON KNOT INVARIANTS AND ITS APPLICATIONS

研究了带状2-结点的有限型不变量或Vassiliev不变量，带状2-结点的HC-移动，链环的HOMFLY多项式的一些性质，保持某些多项式不变量的缠绕运算，以及手铐图的有限型不变量，定义了一类带状2-结点的有限型不变量.然后证明了带状2-结点的归一化Alexander多项式的泰勒展开式中的每个系数都是Vassiliev不变量。在那里，我们用两种方式构建了一个类似瓦西里耶夫的过滤。然而，我们证明了这两个滤子是相同的，因此，这两个有限类型不变量是重合的。我们将HC-Move定义为带状2-节点的解结操作，将其定义为1-节点的Δ-Move的推广。然后给出了带状2-结点的HC-Move与α_2-不变量之间的一些关系，它是2阶有限类型不变量。这使得我们可以决定一些带状2-结的HC-解开数目。利用由于Satoh的带状2-结的虚拟圆弧表示，我们看到HC-Move对应于一个“禁止的动作”，它解开了每个虚拟的结点。然后：(1)我们证明了任何虚拟的结都可以通过禁止的动作来解开。(2)证明了HC-Move是对带状2-结点的虚拟圆弧表示的一种解结操作。(3)给出了1-结点的Δ-Move和自旋2-Not的HC-Move之间的一些关系，给出了用链数和真子链的Homfly多项式的系数多项式描述的链环的Homfly多项式的二、三系数多项式的公式，并介绍了一些关于缠绕二重项的缠绕运算。如果缠结满足某些条件，则得到的链接具有与原始链接相同的多项式不变量。

We studied on finite type invariants or Vassiliev invariants of ribbon 2-knots, HC-moves for ribbon 2-knots, some properties of HOMFLY polynomials of links, tangle surgeries preserving some polynomial invariants, and the finite type invariants for handcuff graphs.We defined finite type invariants for a class of ribbon 2-knots. Then we showed that each coefficient in the Taylor expansion of the normalized Alexander polynomial of a ribbon 2-knot is a Vassiliev invariant. There, we constructed a 'Vassiliev-like' filtration in two ways. However, we proved that the two filtrations are the same, and thus, the two finite type invariants are coincident.We defined the HC-move as an unknotting operation of a ribbon 2-knot as a generalization of a Δ-move for a 1-knot. Then we gave some relatins between the HC-move and the α_2-invariant of a ribbon 2-knot, which is the order 2 finite type invariant. This allowed us to decide the HC-unknotting numbers of some ribbon 2-konts.Making use of the virtual arc representation of a ribbon 2-knot due to Satoh, we saw that the HC-move corresponds to one of the "forbidden moves", which unknot every virtual knot. Then : (1) We proved that any virtural knot can be unknotted by the forbidden moves. (2) We proved the HC-move is an unknotting operation for the virtual arc representation of a ribbon 2-knot. (3) We gave some relation between the Δ-move for a 1-knot and the HC-move for the spun 2-knot.We give formulas for the second and third coefficient polynomials of the HOMFLY polynomial of a link which are described by the linking numbers and the coefficient polynomials of the HOMFLY polynomials of the proper sublinks.We introduce some tangle surgeries on the double of a tangle. If the tangle satisfies certain conditions, then the resulting link has the same polynomial invariant as the original one.