Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2018-02-28 DOI:10.4171/qt/133
C. Leverson, Dan Rutherford
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引用次数: 8

Abstract

We establish relationships between two classes of invariants of Legendrian knots in $\mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $\beta \subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R^m_{S(K,\beta)}(z)$ of the satellite of $K$ with $\beta$ specialized at $z=q^{1/2}-q^{-1/2}$ with $q$ a prime power, and we use this formula to prove that arbitrary $m$-graded satellite ruling polynomials, $R^m_{S(K,L)}$, are determined by the Chekanov-Eliashberg DGA of $K$. Conversely, for $m\neq 1$, we introduce an $n$-colored $m$-graded ruling polynomial, $R^m_{n,K}(q)$, in strict analogy with the $n$-colored HOMFLY-PT polynomial, and show that the total $n$-dimensional $m$-graded representation number of $K$ to $\mathbb{F}_q^n$, $\mbox{Rep}_m(K,\mathbb{F}_q^n)$, is exactly equal to $R^m_{n,K}(q)$. In the case of $2$-graded representations, we show that $R^2_{n,K}=\mbox{Rep}_2(K, \mathbb{F}_q^n)$ arises as a specialization of the $n$-colored HOMFLY-PT polynomial.
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卫星统治多项式,DGA表示,和彩色HOMFLY-PT多项式
建立了$\mathbb{R}^3$中两类Legendrian节的不变量之间的关系:Chekanov-Eliashberg DGA和卫星统治多项式的表示数。对于正排列辫$\beta \subset J^1S^1$,我们给出了$K$卫星的$m$分级统治多项式$R^m_{S(K,\beta)}(z)$的表示数的精确公式,其中$\beta$专为$z=q^{1/2}-q^{-1/2}$, $q$是一个素数幂,我们用这个公式证明了任意$m$分级卫星统治多项式$R^m_{S(K,L)}$是由$K$的Chekanov-Eliashberg DGA确定的。相反,对于$m\neq 1$,我们引入了一个$n$ -colored $m$ -graded的统治多项式$R^m_{n,K}(q)$,与$n$ -colored HOMFLY-PT多项式严格类比,并证明了$K$到$\mathbb{F}_q^n$、$\mbox{Rep}_m(K,\mathbb{F}_q^n)$的$n$ -dimensional $m$ -graded的总表示数正好等于$R^m_{n,K}(q)$。在$2$ -分级表示的情况下,我们表明$R^2_{n,K}=\mbox{Rep}_2(K, \mathbb{F}_q^n)$是$n$ -彩色HOMFLY-PT多项式的专一化。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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