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STUDY ON KNOT INVARIANTS AND ITS APPLICATIONS

结不变量的研究及其应用

基本信息

批准号：
11640090
负责人：
KANENOBU Taizo
金额：
$ 2.24万
依托单位：
Osaka City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

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.

我们研究了带状 2 结的有限类型不变量或 Vassiliev 不变量、带状 2 结的 HC 移动、链接 HOMFLY 多项式的一些性质、保留一些多项式不变量的缠结手术以及手铐图的有限类型不变量。我们为一类带状 2 结定义了有限类型不变量。然后我们证明带状 2 结的归一化亚历山大多项式的泰勒展开式中的每个系数都是瓦西里耶夫不变量。在那里，我们通过两种方式构建了“类似瓦西里耶夫”的过滤。然而，我们证明了这两个过滤是相同的，因此，两个有限类型不变量是一致的。我们将 HC 移动定义为带状 2 结的解结操作，作为 1 结的 Δ 移动的推广。然后我们给出了 HC 移动和带状 2 结的 α_2 不变量（2 阶有限类型不变量）之间的一些关系。这使我们能够确定一些丝带 2 结的 HC 解结数。利用 Satoh 提出的丝带 2 结的虚拟弧表示，我们看到 HC 移动对应于“禁止移动”之一，它解开每个虚拟结。然后：（1）我们证明了任何虚拟结都可以通过禁止的动作来解开。 (2) 我们证明了 HC-move 是对带状 2 结的虚拟弧表示的解结操作。 (3) 我们给出了 1 结的 Δ 移动和旋转 2 结的 HC 移动之间的一些关系。我们给出了链接 HOMFLY 多项式的第二和第三系数多项式的公式，这些公式由链接数和真子链接的 HOMFLY 多项式的系数多项式描述。我们介绍了一些关于缠结的双重的缠结手术。如果缠结满足某些条件，则生成的链接具有与原始链接相同的多项式不变量。