Examples on polynomial invariants of knots and links
Examples on polynomial invariants of knots and links
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结和链接的多项式不变量示例
DOI：
10.1007/bf01459137
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发表时间：
1986
影响因子：
1.4
通讯作者：
T. Kanenobu
中科院分区：
文献类型：
--
作者：
T. Kanenobu
1984年，Jones[12]发现了三维球面上从辫子群表示到某些有限维von Neumann代数的定向纽结或环的同构型多项式不变量。琼斯多项式也是通过一个递归公式计算的，该递归公式涉及改变结点或链节的投影中的交叉点，直到得到解链为止。从那时起，一些新的多项式不变量被发现。直接推广Jones的方法或使用递归公式，Freyd等人[7]，Przytycki和Traczyk[26]几乎同时独立地发现了定向纽结或环的等价型的二变量Laurent多项式不变量。这种二元琼斯多项式既适用于琼斯多项式，也适用于经典的简化亚历山大多项式。后者也是由多个有价值的亚历山大多项式所产生的一个环。采用更一般的递归公式，Brandt等人[2]和Ho[10]发现了无向纽结或环的一个多项式不变量，称为Q或绝对多项式，见第2节。最近Kauffman[15]发现了一个专门用于Jones多项式和Q多项式的二变量多项式不变量[17]。本文的目的是通过几个例子来考虑这些多项式不变量的异同，除了考夫曼多项式外，二元琼斯多项式，因此，琼斯多项式和约化的亚历山大多项式是倾斜不变的，见第1节，但不是Q多项式。在第四节中，我们考虑一族带状纽结K(a，b)，它是作者在[14]中给出的纽结族的推广。这个族可以完全归类到两个变量的Jones多项式或Jones和Alexander多项式的Skein等价性。每个Skein等价类包含无穷多个纽结，特别是K(a+1，-a)，a=0，1，2，…，它们都是Skein等价的，完全由q多项式来分类(定理3)。由Jones和Bman提出的Jones多项式(推论1.1或1.2)的V~公式与简化的Alexander多项式相比是一个特殊的(或在某种意义上是弱的)点。在第5节中，使用这个公式，我们给出了一个例子
In 1984, Jones [12] discovered a polynomial invariant of the isotopy type of an oriented knot or link in a 3-sphere from a representation of the braid group into certain finite dimensional von Neumann algebras. The Jones polynomial is also calculated by a recursive formula involving changing crossings in a projection of the knot or link until the unlink is obtained. Since then, some new polynomial invariants have been discovered. Generalizing Jones' method directly or using a recursive formula, Freyd et al.[7], and Przytycki and Traczyk [26] discovered at almost the same time independently a 2-variable Laurent polynomial invariant of the isotopy type of an oriented knot or link. This 2-variable Jones polynomial specializes to both the Jones and the classical reduced Alexander polynomials. The latter is also produced by the many valuable Alexander polynomial of a link. Adopting a more general recursive formula, Brandt et al.[2] and Ho [10] discovered a polynomial invariant of an unoriented knot or link, which is called the Q or absolute polynomial, see Sect. 2. And most recently Kauffman [15] discovered a 2-variable polynomial invariant which specializes to both the Jones polynomial and the Q polynomial [17]. The purpose of this paper is to consider the difference and the similarity of these polynomial invariants except for the Kauffman polynomial through several examples.The 2-variable Jones polynomial, and therefore, the Jones and the reduced Alexander polynomials are skein invariant, see Sect. 1, but not the Q polynomial. In Sect. 4, we consider a family of ribbon knots K (a, b), which is a generalization of the family of knots given by the author in [14]. This family can be completely classified up to skein equivalence by either the 2-variable Jones polynomial or the Jones and the Alexander polynomials. Each of the skein equivalent class contains infinitely many knots, and especially the knots K (a+ 1,-a), a= 0, 1, 2,..., which are all skein equivalent, are completely classified by the Q polynomial (Theorem 3). The V~ formula for the Jones polynomial (Corollaries 1.1 or 1.2) devised by Jones and Birman, is a special (or weak, in a sense) point compared with the reduced Alexander polynomial. In Sect. 5, using this formula, we give an example of
纽结理论
DOI：
--
发表时间：
2023
期刊：
影响因子：
--
作者：
Kouki Taniyama;小川竜;Sako Akifumi;谷山 公規
通讯作者：
谷山 公規