Cluster algebras and Jones polynomials
Cluster algebras and Jones polynomials
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簇代数和琼斯多项式
DOI：
10.1007/s00029-019-0503-x
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Schiffler, Ralf
中科院分区：
文献类型：
--
作者：
Lee, Kyungyong;Schiffler, Ralf
我们提出了一个新的和非常具体的集群代数和纽结理论之间的联系。这种联系是通过连分数和蛇图建立的。众所周知，2-桥结和链接类是由连分数参数化的，最近研究表明，可以将每个连分数与蛇图关联起来，从而将其与簇代数中的簇变量关联起来。我们表明，到规范化的领导长期的琼斯多项式的2桥链接是等于专业化的集群变量通过设置所有初始集群变量为1和专业化的初始主系数的集群代数如下，为所有。作为结果，我们得到了一个直接的公式琼斯多项式的2桥链接的分子连续分数的洛朗多项式。我们还得到了琼斯多项式的次数和宽度的公式，以及前三个和后三个系数的公式。沿着这条路，我们还发展了一些关于偶连分数的基本事实，并构造了偶连分数的蛇图。证明了当连分数的值相同时，偶数连分数的蛇图与正连分数的蛇图同构。给出了琼斯多项式的递推公式。
We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by continued fractions, and it has recently been shown that one can associate to each continued fraction a snake graph, and hence a cluster variable in a cluster algebra. We show that up to normalization by the leading term the Jones polynomial of the 2-bridge link is equal to the specialization of this cluster variable obtained by setting all initial cluster variables to 1 and specializing the initial principal coefficients of the cluster algebra as followsand, for all. As a consequence we obtain a direct formula for the Jones polynomial of a 2-bridge link as the numerator of a continued fraction of Laurent polynomials in. We also obtain formulas for the degree and the width of the Jones polynomial, as well as for the first three and the last three coefficients. Along the way, we also develop some basic facts about even continued fractions and construct their snake graphs. We show that the snake graph of an even continued fraction is isomorphic to the snake graph of a positive continued fraction if the continued fractions have the same value. We also give recursive formulas for the Jones polynomials.
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DOI：
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