A Knot Invariant Defined Based on the Skein Relation with Two Equations

IF 4.6 2区 数学 Q1 MATHEMATICS, APPLIED Applied and Computational Mathematics Pub Date : 2021-10-16 DOI:10.11648/J.ACM.20211005.12
Liu Weili, Huimin Lu
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Abstract

Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the fk(L), the property of f(L) is obtained by using the properties of fk(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.
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基于两方程绞合关系的一个结不变量
绳结理论是几何拓扑学的一个分支,绳结理论的核心问题是探讨绳结的等价分类;换句话说,对于一个结,如何判断这个结是否为解结;给定两个结,如何确定这两个结是否相等。要证明两个结是等价的,需要通过相同的标记变换把一个结变成另一个结,但要证明两个结是不等的,问题并不像人们想象的那么简单。我们不能说它们是不等的,因为我们看不到它们之间的变形。对于结点的等价分类问题,我们主要寻找结点之间的等价不变量。目前,学者们也对多个结不变量进行了定义,但也存在一定的局限性,更加难以理解。本文在已有理论结果的基础上,通过两个方程的绞结关系定义了一个结不变量。为了证明这个结不变量,我们定义了一个函数f(L),为了证明f(L)是一个非有向连杆的同调不变量,我们需要证明它在Reideminster运动下保持不变。本文首先定义了fk(L),利用fk(L)的性质得到了f(L)的性质。在证明过程中,多次使用了归纳法。证明过程有些复杂,但比较容易理解。一般的结点不变量用一个方程来定义,本文用两个方程来定义结点不变量。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
8.80
自引率
5.00%
发文量
18
审稿时长
6 months
期刊介绍: Applied and Computational Mathematics (ISSN Online: 2328-5613, ISSN Print: 2328-5605) is a prestigious journal that focuses on the field of applied and computational mathematics. It is driven by the computational revolution and places a strong emphasis on innovative applied mathematics with potential for real-world applicability and practicality. The journal caters to a broad audience of applied mathematicians and scientists who are interested in the advancement of mathematical principles and practical aspects of computational mathematics. Researchers from various disciplines can benefit from the diverse range of topics covered in ACM. To ensure the publication of high-quality content, all research articles undergo a rigorous peer review process. This process includes an initial screening by the editors and anonymous evaluation by expert reviewers. This guarantees that only the most valuable and accurate research is published in ACM.
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