STUDY ON KNOT INVARIANTS AND ITS APPLICATIONS

我们研究了带状2节的有限型不变量或Vassiliev不变量，带状2节的hc -move，连杆HOMFLY多项式的一些性质，保留多项式不变量的缠结手术，以及手铐图的有限型不变量。我们定义了一类带状2节的有限型不变量。然后我们证明了带状2结的标准化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，我们以两种方式构建了“Vassiliev-like”过滤。然而，我们证明了这两个过滤是相同的，因此，这两个有限型不变量是重合的。我们将HC-move定义为带状2结的解结操作，作为1结的Δ-move的概括。然后给出了HC-move与带状2-结α_2-不变量（2阶有限型不变量）之间的关系。这使我们能够决定一些2孔织带的hc解结数。利用佐藤的带状2结的虚拟弧表示，我们看到hc移动对应于一个“禁止移动”，它解开每个虚拟结。然后：(1)证明了任何虚结都可以用禁招解结。(2)我们证明了HC-move是带状2结的虚拟弧表示的解结操作。(3)我们给出了1-结的Δ-move和纺丝2-结的HC-move之间的一些关系。给出了连杆的HOMFLY多项式的第二和第三系数多项式的表达式，用连杆数和适当子连杆的HOMFLY多项式的系数多项式来描述。我们介绍一些缠结手术对缠结的双重。如果缠结满足一定条件，则生成的链路与原始链路具有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.